:: TOPREAL5 semantic presentation

Lemma1: for b1, b2, b3 being real number holds
( b3 in [.b1,b2.] iff ( b1 <= b3 & b3 <= b2 ) )
by RCOMP_1:48;

Lemma2: for b1, b2, b3 being non empty TopSpace
for b4 being continuous Function of b1,b2
for b5 being continuous Function of b2,b3 holds
b5 * b4 is continuous Function of b1,b3
by TOPS_2:58;

theorem Th1: :: TOPREAL5:1
canceled;

theorem Th2: :: TOPREAL5:2
canceled;

theorem Th3: :: TOPREAL5:3
canceled;

theorem Th4: :: TOPREAL5:4
for b1 being non empty TopSpace
for b2, b3, b4 being Subset of b1 holds
not ( b3 is open & b4 is open & b3 meets b2 & b4 meets b2 & b2 c= b3 \/ b4 & b3 misses b4 & b2 is connected )
proof end;

theorem Th5: :: TOPREAL5:5
for b1, b2 being non empty TopSpace
for b3 being continuous Function of b1,b2
for b4 being Subset of b1 holds
( b4 is connected implies b3 .: b4 is connected ) by TOPS_2:75;

theorem Th6: :: TOPREAL5:6
for b1, b2 being real number holds
( b1 <= b2 implies [#] (Closed-Interval-TSpace b1,b2) is connected )
proof end;

Lemma6: for b1 being Subset of R^1
for b2 being real number holds
( b1 = { b3 where B is Element of REAL : b2 < b3 } implies b1 is open )
by JORDAN2B:31;

Lemma7: for b1 being Subset of R^1
for b2 being real number holds
( b1 = { b3 where B is Element of REAL : b2 > b3 } implies b1 is open )
by JORDAN2B:30;

theorem Th7: :: TOPREAL5:7
canceled;

theorem Th8: :: TOPREAL5:8
canceled;

theorem Th9: :: TOPREAL5:9
for b1 being Subset of R^1
for b2 being real number holds
not ( not b2 in b1 & ex b3, b4 being real number st
( b3 in b1 & b4 in b1 & b3 < b2 & b2 < b4 ) & b1 is connected )
proof end;

theorem Th10: :: TOPREAL5:10
for b1 being non empty TopSpace
for b2, b3 being Point of b1
for b4, b5, b6 being Real
for b7 being continuous Function of b1,R^1 holds
not ( b1 is connected & b7 . b2 = b4 & b7 . b3 = b5 & b4 <= b6 & b6 <= b5 & ( for b8 being Point of b1 holds
not b7 . b8 = b6 ) )
proof end;

theorem Th11: :: TOPREAL5:11
for b1 being non empty TopSpace
for b2, b3 being Point of b1
for b4 being Subset of b1
for b5, b6, b7 being Real
for b8 being continuous Function of b1,R^1 holds
not ( b4 is connected & b8 . b2 = b5 & b8 . b3 = b6 & b5 <= b7 & b7 <= b6 & b2 in b4 & b3 in b4 & ( for b9 being Point of b1 holds
not ( b9 in b4 & b8 . b9 = b7 ) ) )
proof end;

theorem Th12: :: TOPREAL5:12
for b1, b2, b3, b4 being real number holds
( b1 < b2 implies for b5 being continuous Function of (Closed-Interval-TSpace b1,b2),R^1
for b6 being real number holds
not ( b5 . b1 = b3 & b5 . b2 = b4 & b3 < b6 & b6 < b4 & ( for b7 being Element of REAL holds
not ( b5 . b7 = b6 & b1 < b7 & b7 < b2 ) ) ) )
proof end;

theorem Th13: :: TOPREAL5:13
for b1, b2, b3, b4 being real number holds
( b1 < b2 implies for b5 being continuous Function of (Closed-Interval-TSpace b1,b2),R^1
for b6 being real number holds
not ( b5 . b1 = b3 & b5 . b2 = b4 & b3 > b6 & b6 > b4 & ( for b7 being Element of REAL holds
not ( b5 . b7 = b6 & b1 < b7 & b7 < b2 ) ) ) )
proof end;

theorem Th14: :: TOPREAL5:14
for b1, b2 being real number
for b3 being continuous Function of (Closed-Interval-TSpace b1,b2),R^1
for b4, b5 being real number holds
not ( b1 < b2 & b4 * b5 < 0 & b4 = b3 . b1 & b5 = b3 . b2 & ( for b6 being Element of REAL holds
not ( b3 . b6 = 0 & b1 < b6 & b6 < b2 ) ) )
proof end;

theorem Th15: :: TOPREAL5:15
for b1 being Function of I[01] ,R^1
for b2, b3 being real number holds
not ( b1 is continuous & b1 . 0 <> b1 . 1 & b2 = b1 . 0 & b3 = b1 . 1 & ( for b4 being Element of REAL holds
not ( 0 < b4 & b4 < 1 & b1 . b4 = (b2 + b3) / 2 ) ) )
proof end;

theorem Th16: :: TOPREAL5:16
for b1 being Function of (TOP-REAL 2),R^1 holds
( b1 = proj1 implies b1 is continuous )
proof end;

theorem Th17: :: TOPREAL5:17
for b1 being Function of (TOP-REAL 2),R^1 holds
( b1 = proj2 implies b1 is continuous )
proof end;

theorem Th18: :: TOPREAL5:18
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Function of I[01] ,((TOP-REAL 2) | b1) holds
not ( b2 is continuous & ( for b3 being Function of I[01] ,R^1 holds
not ( b3 is continuous & ( for b4 being Element of REAL
for b5 being Point of (TOP-REAL 2) holds
( b4 in the carrier of I[01] & b5 = b2 . b4 implies b5 `1 = b3 . b4 ) ) ) ) )
proof end;

theorem Th19: :: TOPREAL5:19
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Function of I[01] ,((TOP-REAL 2) | b1) holds
not ( b2 is continuous & ( for b3 being Function of I[01] ,R^1 holds
not ( b3 is continuous & ( for b4 being Element of REAL
for b5 being Point of (TOP-REAL 2) holds
( b4 in the carrier of I[01] & b5 = b2 . b4 implies b5 `2 = b3 . b4 ) ) ) ) )
proof end;

theorem Th20: :: TOPREAL5:20
for b1 being non empty Subset of (TOP-REAL 2) holds
( b1 is being_simple_closed_curve implies for b2 being Element of REAL holds
ex b3 being Point of (TOP-REAL 2) st
( b3 in b1 & not b3 `2 = b2 ) )
proof end;

theorem Th21: :: TOPREAL5:21
for b1 being non empty Subset of (TOP-REAL 2) holds
( b1 is being_simple_closed_curve implies for b2 being Element of REAL holds
ex b3 being Point of (TOP-REAL 2) st
( b3 in b1 & not b3 `1 = b2 ) )
proof end;

theorem Th22: :: TOPREAL5:22
for b1 being non empty compact Subset of (TOP-REAL 2) holds
not ( b1 is_simple_closed_curve & not N-bound b1 > S-bound b1 )
proof end;

theorem Th23: :: TOPREAL5:23
for b1 being non empty compact Subset of (TOP-REAL 2) holds
not ( b1 is_simple_closed_curve & not E-bound b1 > W-bound b1 )
proof end;

theorem Th24: :: TOPREAL5:24
for b1 being non empty compact Subset of (TOP-REAL 2) holds
not ( b1 is_simple_closed_curve & not S-min b1 <> N-max b1 )
proof end;

theorem Th25: :: TOPREAL5:25
for b1 being non empty compact Subset of (TOP-REAL 2) holds
not ( b1 is_simple_closed_curve & not W-min b1 <> E-max b1 )
proof end;

registration
cluster -> non horizontal non vertical Element of K22(the carrier of (TOP-REAL 2));
coherence
for b1 being Simple_closed_curve holds
( not b1 is vertical & not b1 is horizontal )
proof end;
end;