:: JGRAPH_2 semantic presentation

Lemma1: TOP-REAL 2 = TopSpaceMetr (Euclid 2)
by EUCLID:def 8;

theorem Th1: :: JGRAPH_2:1
canceled;

theorem Th2: :: JGRAPH_2:2
for b1 being real number holds
( 1 <= b1 implies b1 <= b1 ^2 ) by REAL_2:146;

theorem Th3: :: JGRAPH_2:3
for b1 being real number holds
( - 1 >= b1 implies - b1 <= b1 ^2 )
proof end;

theorem Th4: :: JGRAPH_2:4
for b1 being real number holds
not ( - 1 > b1 & not - b1 < b1 ^2 )
proof end;

theorem Th5: :: JGRAPH_2:5
for b1, b2 being real number holds
( b2 ^2 <= b1 ^2 & b1 >= 0 implies ( - b1 <= b2 & b2 <= b1 ) )
proof end;

theorem Th6: :: JGRAPH_2:6
for b1, b2 being real number holds
( b2 ^2 < b1 ^2 & b1 >= 0 implies ( - b1 < b2 & b2 < b1 ) )
proof end;

theorem Th7: :: JGRAPH_2:7
for b1, b2 being real number holds
( - b1 <= b2 & b2 <= b1 implies b2 ^2 <= b1 ^2 )
proof end;

theorem Th8: :: JGRAPH_2:8
for b1, b2 being real number holds
not ( - b1 < b2 & b2 < b1 & not b2 ^2 < b1 ^2 )
proof end;

theorem Th9: :: JGRAPH_2:9
for b1, b2, b3, b4 being non empty TopSpace
for b5 being Function of b1,b2
for b6 being Function of b3,b2
for b7, b8 being Subset of b4 holds
not ( b1 is SubSpace of b4 & b3 is SubSpace of b4 & b7 = [#] b1 & b8 = [#] b3 & ([#] b1) \/ ([#] b3) = [#] b4 & b7 is closed & b8 is closed & b5 is continuous & b6 is continuous & ( for b9 being set holds
( b9 in ([#] b1) /\ ([#] b3) implies b5 . b9 = b6 . b9 ) ) & ( for b9 being Function of b4,b2 holds
not ( b9 = b5 +* b6 & b9 is continuous ) ) )
proof end;

theorem Th10: :: JGRAPH_2:10
for b1 being Nat
for b2 being Point of (Euclid b1)
for b3 being Point of (TOP-REAL b1)
for b4 being real number holds
( b3 = b2 implies Ball b2,b4 = { b5 where B is Point of (TOP-REAL b1) : |.(b3 - b5).| < b4 } )
proof end;

theorem Th11: :: JGRAPH_2:11
( (0.REAL 2) `1 = 0 & (0.REAL 2) `2 = 0 ) by EUCLID:56, EUCLID:58;

theorem Th12: :: JGRAPH_2:12
1.REAL 2 = <*1,1*>
proof end;

theorem Th13: :: JGRAPH_2:13
( (1.REAL 2) `1 = 1 & (1.REAL 2) `2 = 1 )
proof end;

theorem Th14: :: JGRAPH_2:14
( dom proj1 = the carrier of (TOP-REAL 2) & dom proj1 = REAL 2 )
proof end;

theorem Th15: :: JGRAPH_2:15
( dom proj2 = the carrier of (TOP-REAL 2) & dom proj2 = REAL 2 )
proof end;

theorem Th16: :: JGRAPH_2:16
canceled;

theorem Th17: :: JGRAPH_2:17
canceled;

theorem Th18: :: JGRAPH_2:18
for b1 being Point of (TOP-REAL 2) holds b1 = |[(proj1 . b1),(proj2 . b1)]|
proof end;

theorem Th19: :: JGRAPH_2:19
for b1 being Subset of (TOP-REAL 2) holds
( b1 = {(0.REAL 2)} implies ( b1 ` <> {} & the carrier of (TOP-REAL 2) \ b1 <> {} ) )
proof end;

theorem Th20: :: JGRAPH_2:20
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( b3 is continuous iff for b4 being Point of b1
for b5 being Subset of b2 holds
not ( b3 . b4 in b5 & b5 is open & ( for b6 being Subset of b1 holds
not ( b4 in b6 & b6 is open & b3 .: b6 c= b5 ) ) ) )
proof end;

theorem Th21: :: JGRAPH_2:21
for b1 being Point of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2) holds
not ( b2 is open & b1 in b2 & ( for b3 being real number holds
not ( b3 > 0 & { b4 where B is Point of (TOP-REAL 2) : ( (b1 `1 ) - b3 < b4 `1 & b4 `1 < (b1 `1 ) + b3 & (b1 `2 ) - b3 < b4 `2 & b4 `2 < (b1 `2 ) + b3 ) } c= b2 ) ) )
proof end;

theorem Th22: :: JGRAPH_2:22
for b1, b2, b3 being non empty TopSpace
for b4 being Subset of b2
for b5 being Subset of b3
for b6 being Function of b1,b2
for b7 being Function of (b2 | b4),(b3 | b5) holds
not ( b6 is continuous & b7 is continuous & rng b6 c= b4 & b4 <> {} & b5 <> {} & ( for b8 being Function of b1,b3 holds
not ( b8 is continuous & b8 = b7 * b6 ) ) )
proof end;

definition
func Out_In_Sq -> Function of the carrier of (TOP-REAL 2) \ {(0.REAL 2)},the carrier of (TOP-REAL 2) \ {(0.REAL 2)} means :Def1: :: JGRAPH_2:def 1
for b1 being Point of (TOP-REAL 2) holds
( b1 <> 0.REAL 2 implies ( ( ( ( b1 `2 <= b1 `1 & - (b1 `1 ) <= b1 `2 ) or ( b1 `2 >= b1 `1 & b1 `2 <= - (b1 `1 ) ) ) implies a1 . b1 = |[(1 / (b1 `1 )),(((b1 `2 ) / (b1 `1 )) / (b1 `1 ))]| ) & not ( not ( b1 `2 <= b1 `1 & - (b1 `1 ) <= b1 `2 ) & not ( b1 `2 >= b1 `1 & b1 `2 <= - (b1 `1 ) ) & not a1 . b1 = |[(((b1 `1 ) / (b1 `2 )) / (b1 `2 )),(1 / (b1 `2 ))]| ) ) );
existence
ex b1 being Function of the carrier of (TOP-REAL 2) \ {(0.REAL 2)},the carrier of (TOP-REAL 2) \ {(0.REAL 2)} st
for b2 being Point of (TOP-REAL 2) holds
( b2 <> 0.REAL 2 implies ( ( ( ( b2 `2 <= b2 `1 & - (b2 `1 ) <= b2 `2 ) or ( b2 `2 >= b2 `1 & b2 `2 <= - (b2 `1 ) ) ) implies b1 . b2 = |[(1 / (b2 `1 )),(((b2 `2 ) / (b2 `1 )) / (b2 `1 ))]| ) & not ( not ( b2 `2 <= b2 `1 & - (b2 `1 ) <= b2 `2 ) & not ( b2 `2 >= b2 `1 & b2 `2 <= - (b2 `1 ) ) & not b1 . b2 = |[(((b2 `1 ) / (b2 `2 )) / (b2 `2 )),(1 / (b2 `2 ))]| ) ) )
proof end;
uniqueness
for b1, b2 being Function of the carrier of (TOP-REAL 2) \ {(0.REAL 2)},the carrier of (TOP-REAL 2) \ {(0.REAL 2)} holds
( ( for b3 being Point of (TOP-REAL 2) holds
( b3 <> 0.REAL 2 implies ( ( ( ( b3 `2 <= b3 `1 & - (b3 `1 ) <= b3 `2 ) or ( b3 `2 >= b3 `1 & b3 `2 <= - (b3 `1 ) ) ) implies b1 . b3 = |[(1 / (b3 `1 )),(((b3 `2 ) / (b3 `1 )) / (b3 `1 ))]| ) & not ( not ( b3 `2 <= b3 `1 & - (b3 `1 ) <= b3 `2 ) & not ( b3 `2 >= b3 `1 & b3 `2 <= - (b3 `1 ) ) & not b1 . b3 = |[(((b3 `1 ) / (b3 `2 )) / (b3 `2 )),(1 / (b3 `2 ))]| ) ) ) ) & ( for b3 being Point of (TOP-REAL 2) holds
( b3 <> 0.REAL 2 implies ( ( ( ( b3 `2 <= b3 `1 & - (b3 `1 ) <= b3 `2 ) or ( b3 `2 >= b3 `1 & b3 `2 <= - (b3 `1 ) ) ) implies b2 . b3 = |[(1 / (b3 `1 )),(((b3 `2 ) / (b3 `1 )) / (b3 `1 ))]| ) & not ( not ( b3 `2 <= b3 `1 & - (b3 `1 ) <= b3 `2 ) & not ( b3 `2 >= b3 `1 & b3 `2 <= - (b3 `1 ) ) & not b2 . b3 = |[(((b3 `1 ) / (b3 `2 )) / (b3 `2 )),(1 / (b3 `2 ))]| ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def1 defines Out_In_Sq JGRAPH_2:def 1 :
for b1 being Function of the carrier of (TOP-REAL 2) \ {(0.REAL 2)},the carrier of (TOP-REAL 2) \ {(0.REAL 2)} holds
( b1 = Out_In_Sq iff for b2 being Point of (TOP-REAL 2) holds
( b2 <> 0.REAL 2 implies ( ( ( ( b2 `2 <= b2 `1 & - (b2 `1 ) <= b2 `2 ) or ( b2 `2 >= b2 `1 & b2 `2 <= - (b2 `1 ) ) ) implies b1 . b2 = |[(1 / (b2 `1 )),(((b2 `2 ) / (b2 `1 )) / (b2 `1 ))]| ) & not ( not ( b2 `2 <= b2 `1 & - (b2 `1 ) <= b2 `2 ) & not ( b2 `2 >= b2 `1 & b2 `2 <= - (b2 `1 ) ) & not b1 . b2 = |[(((b2 `1 ) / (b2 `2 )) / (b2 `2 )),(1 / (b2 `2 ))]| ) ) ) );

theorem Th23: :: JGRAPH_2:23
for b1 being Point of (TOP-REAL 2) holds
not ( not ( b1 `2 <= b1 `1 & - (b1 `1 ) <= b1 `2 ) & not ( b1 `2 >= b1 `1 & b1 `2 <= - (b1 `1 ) ) & not ( b1 `1 <= b1 `2 & - (b1 `2 ) <= b1 `1 ) & not ( b1 `1 >= b1 `2 & b1 `1 <= - (b1 `2 ) ) )
proof end;

theorem Th24: :: JGRAPH_2:24
for b1 being Point of (TOP-REAL 2) holds
( b1 <> 0.REAL 2 implies ( ( ( ( b1 `1 <= b1 `2 & - (b1 `2 ) <= b1 `1 ) or ( b1 `1 >= b1 `2 & b1 `1 <= - (b1 `2 ) ) ) implies Out_In_Sq . b1 = |[(((b1 `1 ) / (b1 `2 )) / (b1 `2 )),(1 / (b1 `2 ))]| ) & not ( not ( b1 `1 <= b1 `2 & - (b1 `2 ) <= b1 `1 ) & not ( b1 `1 >= b1 `2 & b1 `1 <= - (b1 `2 ) ) & not Out_In_Sq . b1 = |[(1 / (b1 `1 )),(((b1 `2 ) / (b1 `1 )) / (b1 `1 ))]| ) ) )
proof end;

theorem Th25: :: JGRAPH_2:25
for b1 being Subset of (TOP-REAL 2)
for b2 being Subset of ((TOP-REAL 2) | b1) holds
( b2 = { b3 where B is Point of (TOP-REAL 2) : ( ( ( b3 `2 <= b3 `1 & - (b3 `1 ) <= b3 `2 ) or ( b3 `2 >= b3 `1 & b3 `2 <= - (b3 `1 ) ) ) & b3 <> 0.REAL 2 ) } implies rng (Out_In_Sq | b2) c= the carrier of (((TOP-REAL 2) | b1) | b2) )
proof end;

theorem Th26: :: JGRAPH_2:26
for b1 being Subset of (TOP-REAL 2)
for b2 being Subset of ((TOP-REAL 2) | b1) holds
( b2 = { b3 where B is Point of (TOP-REAL 2) : ( ( ( b3 `1 <= b3 `2 & - (b3 `2 ) <= b3 `1 ) or ( b3 `1 >= b3 `2 & b3 `1 <= - (b3 `2 ) ) ) & b3 <> 0.REAL 2 ) } implies rng (Out_In_Sq | b2) c= the carrier of (((TOP-REAL 2) | b1) | b2) )
proof end;

theorem Th27: :: JGRAPH_2:27
for b1 being set
for b2 being non empty Subset of (TOP-REAL 2) holds
( b1 = { b3 where B is Point of (TOP-REAL 2) : ( ( ( b3 `2 <= b3 `1 & - (b3 `1 ) <= b3 `2 ) or ( b3 `2 >= b3 `1 & b3 `2 <= - (b3 `1 ) ) ) & b3 <> 0.REAL 2 ) } & b2 ` = {(0.REAL 2)} implies ( b1 is non empty Subset of ((TOP-REAL 2) | b2) & b1 is non empty Subset of (TOP-REAL 2) ) )
proof end;

theorem Th28: :: JGRAPH_2:28
for b1 being set
for b2 being non empty Subset of (TOP-REAL 2) holds
( b1 = { b3 where B is Point of (TOP-REAL 2) : ( ( ( b3 `1 <= b3 `2 & - (b3 `2 ) <= b3 `1 ) or ( b3 `1 >= b3 `2 & b3 `1 <= - (b3 `2 ) ) ) & b3 <> 0.REAL 2 ) } & b2 ` = {(0.REAL 2)} implies ( b1 is non empty Subset of ((TOP-REAL 2) | b2) & b1 is non empty Subset of (TOP-REAL 2) ) )
proof end;

theorem Th29: :: JGRAPH_2:29
for b1 being non empty TopSpace
for b2, b3 being Function of b1,R^1 holds
not ( b2 is continuous & b3 is continuous & ( for b4 being Function of b1,R^1 holds
not ( ( for b5 being Point of b1
for b6, b7 being real number holds
( b2 . b5 = b6 & b3 . b5 = b7 implies b4 . b5 = b6 + b7 ) ) & b4 is continuous ) ) )
proof end;

theorem Th30: :: JGRAPH_2:30
for b1 being non empty TopSpace
for b2 being real number holds
ex b3 being Function of b1,R^1 st
( ( for b4 being Point of b1 holds b3 . b4 = b2 ) & b3 is continuous )
proof end;

theorem Th31: :: JGRAPH_2:31
for b1 being non empty TopSpace
for b2, b3 being Function of b1,R^1 holds
not ( b2 is continuous & b3 is continuous & ( for b4 being Function of b1,R^1 holds
not ( ( for b5 being Point of b1
for b6, b7 being real number holds
( b2 . b5 = b6 & b3 . b5 = b7 implies b4 . b5 = b6 - b7 ) ) & b4 is continuous ) ) )
proof end;

theorem Th32: :: JGRAPH_2:32
for b1 being non empty TopSpace
for b2 being Function of b1,R^1 holds
not ( b2 is continuous & ( for b3 being Function of b1,R^1 holds
not ( ( for b4 being Point of b1
for b5 being real number holds
( b2 . b4 = b5 implies b3 . b4 = b5 * b5 ) ) & b3 is continuous ) ) )
proof end;

theorem Th33: :: JGRAPH_2:33
for b1 being non empty TopSpace
for b2 being Function of b1,R^1
for b3 being real number holds
not ( b2 is continuous & ( for b4 being Function of b1,R^1 holds
not ( ( for b5 being Point of b1
for b6 being real number holds
( b2 . b5 = b6 implies b4 . b5 = b3 * b6 ) ) & b4 is continuous ) ) )
proof end;

theorem Th34: :: JGRAPH_2:34
for b1 being non empty TopSpace
for b2 being Function of b1,R^1
for b3 being real number holds
not ( b2 is continuous & ( for b4 being Function of b1,R^1 holds
not ( ( for b5 being Point of b1
for b6 being real number holds
( b2 . b5 = b6 implies b4 . b5 = b6 + b3 ) ) & b4 is continuous ) ) )
proof end;

theorem Th35: :: JGRAPH_2:35
for b1 being non empty TopSpace
for b2, b3 being Function of b1,R^1 holds
not ( b2 is continuous & b3 is continuous & ( for b4 being Function of b1,R^1 holds
not ( ( for b5 being Point of b1
for b6, b7 being real number holds
( b2 . b5 = b6 & b3 . b5 = b7 implies b4 . b5 = b6 * b7 ) ) & b4 is continuous ) ) )
proof end;

theorem Th36: :: JGRAPH_2:36
for b1 being non empty TopSpace
for b2 being Function of b1,R^1 holds
not ( b2 is continuous & ( for b3 being Point of b1 holds
b2 . b3 <> 0 ) & ( for b3 being Function of b1,R^1 holds
not ( ( for b4 being Point of b1
for b5 being real number holds
( b2 . b4 = b5 implies b3 . b4 = 1 / b5 ) ) & b3 is continuous ) ) )
proof end;

theorem Th37: :: JGRAPH_2:37
for b1 being non empty TopSpace
for b2, b3 being Function of b1,R^1 holds
not ( b2 is continuous & b3 is continuous & ( for b4 being Point of b1 holds
b3 . b4 <> 0 ) & ( for b4 being Function of b1,R^1 holds
not ( ( for b5 being Point of b1
for b6, b7 being real number holds
( b2 . b5 = b6 & b3 . b5 = b7 implies b4 . b5 = b6 / b7 ) ) & b4 is continuous ) ) )
proof end;

theorem Th38: :: JGRAPH_2:38
for b1 being non empty TopSpace
for b2, b3 being Function of b1,R^1 holds
not ( b2 is continuous & b3 is continuous & ( for b4 being Point of b1 holds
b3 . b4 <> 0 ) & ( for b4 being Function of b1,R^1 holds
not ( ( for b5 being Point of b1
for b6, b7 being real number holds
( b2 . b5 = b6 & b3 . b5 = b7 implies b4 . b5 = (b6 / b7) / b7 ) ) & b4 is continuous ) ) )
proof end;

theorem Th39: :: JGRAPH_2:39
for b1 being Subset of (TOP-REAL 2)
for b2 being Function of ((TOP-REAL 2) | b1),R^1 holds
( ( for b3 being Point of ((TOP-REAL 2) | b1) holds b2 . b3 = proj1 . b3 ) implies b2 is continuous )
proof end;

theorem Th40: :: JGRAPH_2:40
for b1 being Subset of (TOP-REAL 2)
for b2 being Function of ((TOP-REAL 2) | b1),R^1 holds
( ( for b3 being Point of ((TOP-REAL 2) | b1) holds b2 . b3 = proj2 . b3 ) implies b2 is continuous )
proof end;

theorem Th41: :: JGRAPH_2:41
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Function of ((TOP-REAL 2) | b1),R^1 holds
( ( for b3 being Point of (TOP-REAL 2) holds
( b3 in the carrier of ((TOP-REAL 2) | b1) implies b2 . b3 = 1 / (b3 `1 ) ) ) & ( for b3 being Point of (TOP-REAL 2) holds
not ( b3 in the carrier of ((TOP-REAL 2) | b1) & not b3 `1 <> 0 ) ) implies b2 is continuous )
proof end;

theorem Th42: :: JGRAPH_2:42
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Function of ((TOP-REAL 2) | b1),R^1 holds
( ( for b3 being Point of (TOP-REAL 2) holds
( b3 in the carrier of ((TOP-REAL 2) | b1) implies b2 . b3 = 1 / (b3 `2 ) ) ) & ( for b3 being Point of (TOP-REAL 2) holds
not ( b3 in the carrier of ((TOP-REAL 2) | b1) & not b3 `2 <> 0 ) ) implies b2 is continuous )
proof end;

theorem Th43: :: JGRAPH_2:43
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Function of ((TOP-REAL 2) | b1),R^1 holds
( ( for b3 being Point of (TOP-REAL 2) holds
( b3 in the carrier of ((TOP-REAL 2) | b1) implies b2 . b3 = ((b3 `2 ) / (b3 `1 )) / (b3 `1 ) ) ) & ( for b3 being Point of (TOP-REAL 2) holds
not ( b3 in the carrier of ((TOP-REAL 2) | b1) & not b3 `1 <> 0 ) ) implies b2 is continuous )
proof end;

theorem Th44: :: JGRAPH_2:44
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Function of ((TOP-REAL 2) | b1),R^1 holds
( ( for b3 being Point of (TOP-REAL 2) holds
( b3 in the carrier of ((TOP-REAL 2) | b1) implies b2 . b3 = ((b3 `1 ) / (b3 `2 )) / (b3 `2 ) ) ) & ( for b3 being Point of (TOP-REAL 2) holds
not ( b3 in the carrier of ((TOP-REAL 2) | b1) & not b3 `2 <> 0 ) ) implies b2 is continuous )
proof end;

theorem Th45: :: JGRAPH_2:45
for b1, b2 being Subset of (TOP-REAL 2)
for b3 being Function of ((TOP-REAL 2) | b1),((TOP-REAL 2) | b2)
for b4, b5 being Function of ((TOP-REAL 2) | b1),R^1 holds
( b4 is continuous & b5 is continuous & b1 <> {} & b2 <> {} & ( for b6, b7, b8, b9 being real number holds
( |[b6,b7]| in b1 & b8 = b4 . |[b6,b7]| & b9 = b5 . |[b6,b7]| implies b3 . |[b6,b7]| = |[b8,b9]| ) ) implies b3 is continuous )
proof end;

theorem Th46: :: JGRAPH_2:46
for b1, b2 being Subset of (TOP-REAL 2)
for b3 being Function of ((TOP-REAL 2) | b1),((TOP-REAL 2) | b2) holds
( b3 = Out_In_Sq | b1 & b2 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & b1 = { b4 where B is Point of (TOP-REAL 2) : ( ( ( b4 `2 <= b4 `1 & - (b4 `1 ) <= b4 `2 ) or ( b4 `2 >= b4 `1 & b4 `2 <= - (b4 `1 ) ) ) & b4 <> 0.REAL 2 ) } implies b3 is continuous )
proof end;

theorem Th47: :: JGRAPH_2:47
for b1, b2 being Subset of (TOP-REAL 2)
for b3 being Function of ((TOP-REAL 2) | b1),((TOP-REAL 2) | b2) holds
( b3 = Out_In_Sq | b1 & b2 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & b1 = { b4 where B is Point of (TOP-REAL 2) : ( ( ( b4 `1 <= b4 `2 & - (b4 `2 ) <= b4 `1 ) or ( b4 `1 >= b4 `2 & b4 `1 <= - (b4 `2 ) ) ) & b4 <> 0.REAL 2 ) } implies b3 is continuous )
proof end;

scheme :: JGRAPH_2:sch 1
s1{ P1[ set ] } :
{ b1 where B is Point of (TOP-REAL 2) : P1[b1] } is Subset of (TOP-REAL 2)
proof end;

scheme :: JGRAPH_2:sch 2
s2{ P1[ set ], F1() -> Subset of (TOP-REAL 2) } :
F1() ` = { b1 where B is Point of (TOP-REAL 2) : not P1[b1] }
provided
E44: F1() = { b1 where B is Point of (TOP-REAL 2) : P1[b1] }
proof end;

E44: now
let c1, c2, c3, c4 be real number ;
set c5 = (c1 - c2) / 4;
assume (c1 - c3) - (c2 - c4) <= ((c1 - c2) / 4) - (- ((c1 - c2) / 4)) ;
then (c1 - c2) - (c3 - c4) <= ((c1 - c2) / 4) + ((c1 - c2) / 4) ;
then c1 - c2 <= (c3 - c4) + (((c1 - c2) / 4) + ((c1 - c2) / 4)) by XREAL_1:22;
then (c1 - c2) - ((c1 - c2) / 2) <= c3 - c4 by XREAL_1:22;
hence (c1 - c2) / 2 <= c3 - c4 ;
end;

scheme :: JGRAPH_2:sch 3
s3{ F1( Point of (TOP-REAL 2)) -> real number , F2( Point of (TOP-REAL 2)) -> real number } :
{ b1 where B is Point of (TOP-REAL 2) : F1(b1) <= F2(b1) } is closed Subset of (TOP-REAL 2)
provided
E45: for b1, b2 being Point of (TOP-REAL 2) holds
( F1((b1 - b2)) = F1(b1) - F1(b2) & F2((b1 - b2)) = F2(b1) - F2(b2) ) and E46: for b1, b2 being Point of (TOP-REAL 2) holds |.(b1 - b2).| ^2 = (F1((b1 - b2)) ^2 ) + (F2((b1 - b2)) ^2 )
proof end;

deffunc H1( Point of (TOP-REAL 2)) -> Element of REAL = a1 `1 ;

deffunc H2( Point of (TOP-REAL 2)) -> Element of REAL = a1 `2 ;

Lemma45: for b1, b2 being Point of (TOP-REAL 2) holds
( H1(b1 - b2) = H1(b1) - H1(b2) & H2(b1 - b2) = H2(b1) - H2(b2) )
by TOPREAL3:8;

Lemma46: for b1, b2 being Point of (TOP-REAL 2) holds |.(b1 - b2).| ^2 = (H1(b1 - b2) ^2 ) + (H2(b1 - b2) ^2 )
by JGRAPH_1:46;

Lemma47: { b1 where B is Point of (TOP-REAL 2) : H1(b1) <= H2(b1) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lemma45, Lemma46);

Lemma48: for b1, b2 being Point of (TOP-REAL 2) holds
( H2(b1 - b2) = H2(b1) - H2(b2) & H1(b1 - b2) = H1(b1) - H1(b2) )
by TOPREAL3:8;

Lemma49: for b1, b2 being Point of (TOP-REAL 2) holds |.(b1 - b2).| ^2 = (H2(b1 - b2) ^2 ) + (H1(b1 - b2) ^2 )
by JGRAPH_1:46;

Lemma50: { b1 where B is Point of (TOP-REAL 2) : H2(b1) <= H1(b1) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lemma48, Lemma49);

deffunc H3( Point of (TOP-REAL 2)) -> Element of REAL = - (a1 `1 );

deffunc H4( Point of (TOP-REAL 2)) -> Element of REAL = - (a1 `2 );

E51: now
let c1, c2 be Point of (TOP-REAL 2);
thus H3(c1 - c2) = - ((c1 `1 ) - (c2 `1 )) by TOPREAL3:8
.= H3(c1) - H3(c2) ;
thus H2(c1 - c2) = H2(c1) - H2(c2) by TOPREAL3:8;
end;

E52: now
let c1, c2 be Point of (TOP-REAL 2);
H3(c1 - c2) ^2 = H1(c1 - c2) ^2 ;
hence |.(c1 - c2).| ^2 = (H3(c1 - c2) ^2 ) + (H2(c1 - c2) ^2 ) by JGRAPH_1:46;
end;

Lemma53: { b1 where B is Point of (TOP-REAL 2) : H3(b1) <= H2(b1) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lemma51, Lemma52);

E54: now
let c1, c2 be Point of (TOP-REAL 2);
thus H2(c1 - c2) = H2(c1) - H2(c2) by TOPREAL3:8;
thus H3(c1 - c2) = - ((c1 `1 ) - (c2 `1 )) by TOPREAL3:8
.= H3(c1) - H3(c2) ;
end;

E55: now
let c1, c2 be Point of (TOP-REAL 2);
(- ((c1 - c2) `1 )) ^2 = ((c1 - c2) `1 ) ^2 ;
hence |.(c1 - c2).| ^2 = (H2(c1 - c2) ^2 ) + (H3(c1 - c2) ^2 ) by JGRAPH_1:46;
end;

Lemma56: { b1 where B is Point of (TOP-REAL 2) : H2(b1) <= H3(b1) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lemma54, Lemma55);

E57: now
let c1, c2 be Point of (TOP-REAL 2);
thus H4(c1 - c2) = - ((c1 `2 ) - (c2 `2 )) by TOPREAL3:8
.= H4(c1) - H4(c2) ;
thus H1(c1 - c2) = H1(c1) - H1(c2) by TOPREAL3:8;
end;

E58: now
let c1, c2 be Point of (TOP-REAL 2);
(- ((c1 - c2) `2 )) ^2 = ((c1 - c2) `2 ) ^2 ;
hence |.(c1 - c2).| ^2 = (H4(c1 - c2) ^2 ) + (H1(c1 - c2) ^2 ) by JGRAPH_1:46;
end;

Lemma59: { b1 where B is Point of (TOP-REAL 2) : H4(b1) <= H1(b1) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lemma57, Lemma58);

E60: now
let c1, c2 be Point of (TOP-REAL 2);
thus H1(c1 - c2) = H1(c1) - H1(c2) by TOPREAL3:8;
thus H4(c1 - c2) = - ((c1 `2 ) - (c2 `2 )) by TOPREAL3:8
.= H4(c1) - H4(c2) ;
end;

E61: now
let c1, c2 be Point of (TOP-REAL 2);
H4(c1 - c2) ^2 = H2(c1 - c2) ^2 ;
hence |.(c1 - c2).| ^2 = (H1(c1 - c2) ^2 ) + (H4(c1 - c2) ^2 ) by JGRAPH_1:46;
end;

Lemma62: { b1 where B is Point of (TOP-REAL 2) : H1(b1) <= H4(b1) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lemma60, Lemma61);

theorem Th48: :: JGRAPH_2:48
for b1 being Subset of (TOP-REAL 2)
for b2 being Subset of ((TOP-REAL 2) | b1)
for b3 being Function of (((TOP-REAL 2) | b1) | b2),((TOP-REAL 2) | b1) holds
( b3 = Out_In_Sq | b2 & b1 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & b2 = { b4 where B is Point of (TOP-REAL 2) : ( ( ( b4 `2 <= b4 `1 & - (b4 `1 ) <= b4 `2 ) or ( b4 `2 >= b4 `1 & b4 `2 <= - (b4 `1 ) ) ) & b4 <> 0.REAL 2 ) } implies ( b3 is continuous & b2 is closed ) )
proof end;

theorem Th49: :: JGRAPH_2:49
for b1 being Subset of (TOP-REAL 2)
for b2 being Subset of ((TOP-REAL 2) | b1)
for b3 being Function of (((TOP-REAL 2) | b1) | b2),((TOP-REAL 2) | b1) holds
( b3 = Out_In_Sq | b2 & b1 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & b2 = { b4 where B is Point of (TOP-REAL 2) : ( ( ( b4 `1 <= b4 `2 & - (b4 `2 ) <= b4 `1 ) or ( b4 `1 >= b4 `2 & b4 `1 <= - (b4 `2 ) ) ) & b4 <> 0.REAL 2 ) } implies ( b3 is continuous & b2 is closed ) )
proof end;

theorem Th50: :: JGRAPH_2:50
for b1 being non empty Subset of (TOP-REAL 2) holds
not ( b1 ` = {(0.REAL 2)} & ( for b2 being Function of ((TOP-REAL 2) | b1),((TOP-REAL 2) | b1) holds
not ( b2 = Out_In_Sq & b2 is continuous ) ) )
proof end;

theorem Th51: :: JGRAPH_2:51
for b1, b2, b3 being Subset of (TOP-REAL 2) holds
not ( b1 = {(0.REAL 2)} & b2 = { b4 where B is Point of (TOP-REAL 2) : ( - 1 < b4 `1 & b4 `1 < 1 & - 1 < b4 `2 & b4 `2 < 1 ) } & b3 = { b4 where B is Point of (TOP-REAL 2) : not ( not ( - 1 = b4 `1 & - 1 <= b4 `2 & b4 `2 <= 1 ) & not ( b4 `1 = 1 & - 1 <= b4 `2 & b4 `2 <= 1 ) & not ( - 1 = b4 `2 & - 1 <= b4 `1 & b4 `1 <= 1 ) & not ( 1 = b4 `2 & - 1 <= b4 `1 & b4 `1 <= 1 ) ) } & ( for b4 being Function of ((TOP-REAL 2) | (b1 ` )),((TOP-REAL 2) | (b1 ` )) holds
not ( b4 is continuous & b4 is one-to-one & ( for b5 being Point of (TOP-REAL 2) holds
not ( b5 in b2 & b5 <> 0.REAL 2 & b4 . b5 in b2 \/ b3 ) ) & ( for b5 being Point of (TOP-REAL 2) holds
( not b5 in b2 \/ b3 implies b4 . b5 in b2 ) ) & ( for b5 being Point of (TOP-REAL 2) holds
( b5 in b3 implies b4 . b5 = b5 ) ) ) ) )
proof end;

theorem Th52: :: JGRAPH_2:52
for b1, b2 being Function of I[01] ,(TOP-REAL 2)
for b3 being Subset of (TOP-REAL 2)
for b4, b5 being Point of I[01] holds
not ( b4 = 0 & b5 = 1 & b1 is continuous & b1 is one-to-one & b2 is continuous & b2 is one-to-one & b3 = { b6 where B is Point of (TOP-REAL 2) : ( - 1 < b6 `1 & b6 `1 < 1 & - 1 < b6 `2 & b6 `2 < 1 ) } & (b1 . b4) `1 = - 1 & (b1 . b5) `1 = 1 & - 1 <= (b1 . b4) `2 & (b1 . b4) `2 <= 1 & - 1 <= (b1 . b5) `2 & (b1 . b5) `2 <= 1 & (b2 . b4) `2 = - 1 & (b2 . b5) `2 = 1 & - 1 <= (b2 . b4) `1 & (b2 . b4) `1 <= 1 & - 1 <= (b2 . b5) `1 & (b2 . b5) `1 <= 1 & rng b1 misses b3 & rng b2 misses b3 & not rng b1 meets rng b2 )
proof end;

theorem Th53: :: JGRAPH_2:53
for b1, b2, b3, b4 being real number
for b5 being Function of (TOP-REAL 2),(TOP-REAL 2) holds
( ( for b6 being Point of (TOP-REAL 2) holds b5 . b6 = |[((b1 * (b6 `1 )) + b2),((b3 * (b6 `2 )) + b4)]| ) implies b5 is continuous )
proof end;

definition
let c1, c2, c3, c4 be real number ;
func AffineMap c1,c2,c3,c4 -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def2: :: JGRAPH_2:def 2
for b1 being Point of (TOP-REAL 2) holds a5 . b1 = |[((a1 * (b1 `1 )) + a2),((a3 * (b1 `2 )) + a4)]|;
existence
ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st
for b2 being Point of (TOP-REAL 2) holds b1 . b2 = |[((c1 * (b2 `1 )) + c2),((c3 * (b2 `2 )) + c4)]|
proof end;
uniqueness
for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) holds
( ( for b3 being Point of (TOP-REAL 2) holds b1 . b3 = |[((c1 * (b3 `1 )) + c2),((c3 * (b3 `2 )) + c4)]| ) & ( for b3 being Point of (TOP-REAL 2) holds b2 . b3 = |[((c1 * (b3 `1 )) + c2),((c3 * (b3 `2 )) + c4)]| ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def2 defines AffineMap JGRAPH_2:def 2 :
for b1, b2, b3, b4 being real number
for b5 being Function of (TOP-REAL 2),(TOP-REAL 2) holds
( b5 = AffineMap b1,b2,b3,b4 iff for b6 being Point of (TOP-REAL 2) holds b5 . b6 = |[((b1 * (b6 `1 )) + b2),((b3 * (b6 `2 )) + b4)]| );

registration
let c1, c2, c3, c4 be real number ;
cluster AffineMap a1,a2,a3,a4 -> continuous ;
coherence
AffineMap c1,c2,c3,c4 is continuous
proof end;
end;

theorem Th54: :: JGRAPH_2:54
for b1, b2, b3, b4 being real number holds
( b1 > 0 & b3 > 0 implies AffineMap b1,b2,b3,b4 is one-to-one )
proof end;

theorem Th55: :: JGRAPH_2:55
for b1, b2 being Function of I[01] ,(TOP-REAL 2)
for b3, b4, b5, b6 being real number
for b7, b8 being Point of I[01] holds
not ( b7 = 0 & b8 = 1 & b1 is continuous & b1 is one-to-one & b2 is continuous & b2 is one-to-one & (b1 . b7) `1 = b3 & (b1 . b8) `1 = b4 & b5 <= (b1 . b7) `2 & (b1 . b7) `2 <= b6 & b5 <= (b1 . b8) `2 & (b1 . b8) `2 <= b6 & (b2 . b7) `2 = b5 & (b2 . b8) `2 = b6 & b3 <= (b2 . b7) `1 & (b2 . b7) `1 <= b4 & b3 <= (b2 . b8) `1 & (b2 . b8) `1 <= b4 & b3 < b4 & b5 < b6 & ( for b9 being Point of I[01] holds
not ( b3 < (b1 . b9) `1 & (b1 . b9) `1 < b4 & b5 < (b1 . b9) `2 & (b1 . b9) `2 < b6 ) ) & ( for b9 being Point of I[01] holds
not ( b3 < (b2 . b9) `1 & (b2 . b9) `1 < b4 & b5 < (b2 . b9) `2 & (b2 . b9) `2 < b6 ) ) & not rng b1 meets rng b2 )
proof end;

theorem Th56: :: JGRAPH_2:56
( { b1 where B is Point of (TOP-REAL 2) : b1 `2 <= b1 `1 } is closed Subset of (TOP-REAL 2) & { b1 where B is Point of (TOP-REAL 2) : b1 `1 <= b1 `2 } is closed Subset of (TOP-REAL 2) ) by Lemma47, Lemma50;

theorem Th57: :: JGRAPH_2:57
( { b1 where B is Point of (TOP-REAL 2) : - (b1 `1 ) <= b1 `2 } is closed Subset of (TOP-REAL 2) & { b1 where B is Point of (TOP-REAL 2) : b1 `2 <= - (b1 `1 ) } is closed Subset of (TOP-REAL 2) ) by Lemma53, Lemma56;

theorem Th58: :: JGRAPH_2:58
( { b1 where B is Point of (TOP-REAL 2) : - (b1 `2 ) <= b1 `1 } is closed Subset of (TOP-REAL 2) & { b1 where B is Point of (TOP-REAL 2) : b1 `1 <= - (b1 `2 ) } is closed Subset of (TOP-REAL 2) ) by Lemma59, Lemma62;