:: BHSP_2 semantic presentation

deffunc H1( RealUnitarySpace) -> Element of the carrier of a1 = 0. a1;

definition
let c1 be RealUnitarySpace;
let c2 be sequence of c1;
attr a2 is convergent means :Def1: :: BHSP_2:def 1
ex b1 being Point of a1 st
for b2 being Real holds
not ( b2 > 0 & ( for b3 being Nat holds
ex b4 being Nat st
( b4 >= b3 & not dist (a2 . b4),b1 < b2 ) ) );
end;

:: deftheorem Def1 defines convergent BHSP_2:def 1 :
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is convergent iff ex b3 being Point of b1 st
for b4 being Real holds
not ( b4 > 0 & ( for b5 being Nat holds
ex b6 being Nat st
( b6 >= b5 & not dist (b2 . b6),b3 < b4 ) ) ) );

theorem Th1: :: BHSP_2:1
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is V7 implies b2 is convergent )
proof end;

theorem Th2: :: BHSP_2:2
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is convergent & ex b4 being Nat st
for b5 being Nat holds
( b4 <= b5 implies b3 . b5 = b2 . b5 ) implies b3 is convergent )
proof end;

theorem Th3: :: BHSP_2:3
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is convergent & b3 is convergent implies b2 + b3 is convergent )
proof end;

theorem Th4: :: BHSP_2:4
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is convergent & b3 is convergent implies b2 - b3 is convergent )
proof end;

theorem Th5: :: BHSP_2:5
for b1 being RealUnitarySpace
for b2 being Real
for b3 being sequence of b1 holds
( b3 is convergent implies b2 * b3 is convergent )
proof end;

theorem Th6: :: BHSP_2:6
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is convergent implies - b2 is convergent )
proof end;

theorem Th7: :: BHSP_2:7
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent implies b3 + b2 is convergent )
proof end;

theorem Th8: :: BHSP_2:8
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent implies b3 - b2 is convergent )
proof end;

theorem Th9: :: BHSP_2:9
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is convergent iff ex b3 being Point of b1 st
for b4 being Real holds
not ( b4 > 0 & ( for b5 being Nat holds
ex b6 being Nat st
( b6 >= b5 & not ||.((b2 . b6) - b3).|| < b4 ) ) ) )
proof end;

definition
let c1 be RealUnitarySpace;
let c2 be sequence of c1;
assume E11: c2 is convergent ;
func lim c2 -> Point of a1 means :Def2: :: BHSP_2:def 2
for b1 being Real holds
not ( b1 > 0 & ( for b2 being Nat holds
ex b3 being Nat st
( b3 >= b2 & not dist (a2 . b3),a3 < b1 ) ) );
existence
ex b1 being Point of c1 st
for b2 being Real holds
not ( b2 > 0 & ( for b3 being Nat holds
ex b4 being Nat st
( b4 >= b3 & not dist (c2 . b4),b1 < b2 ) ) )
by E11, Def1;
uniqueness
for b1, b2 being Point of c1 holds
( ( for b3 being Real holds
not ( b3 > 0 & ( for b4 being Nat holds
ex b5 being Nat st
( b5 >= b4 & not dist (c2 . b5),b1 < b3 ) ) ) ) & ( for b3 being Real holds
not ( b3 > 0 & ( for b4 being Nat holds
ex b5 being Nat st
( b5 >= b4 & not dist (c2 . b5),b2 < b3 ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def2 defines lim BHSP_2:def 2 :
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is convergent implies for b3 being Point of b1 holds
( b3 = lim b2 iff for b4 being Real holds
not ( b4 > 0 & ( for b5 being Nat holds
ex b6 being Nat st
( b6 >= b5 & not dist (b2 . b6),b3 < b4 ) ) ) ) );

theorem Th10: :: BHSP_2:10
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is V7 & b2 in rng b3 implies lim b3 = b2 )
proof end;

theorem Th11: :: BHSP_2:11
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is V7 & ex b4 being Nat st b3 . b4 = b2 implies lim b3 = b2 )
proof end;

theorem Th12: :: BHSP_2:12
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is convergent & ex b4 being Nat st
for b5 being Nat holds
( b5 >= b4 implies b3 . b5 = b2 . b5 ) implies lim b2 = lim b3 )
proof end;

theorem Th13: :: BHSP_2:13
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is convergent & b3 is convergent implies lim (b2 + b3) = (lim b2) + (lim b3) )
proof end;

theorem Th14: :: BHSP_2:14
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is convergent & b3 is convergent implies lim (b2 - b3) = (lim b2) - (lim b3) )
proof end;

theorem Th15: :: BHSP_2:15
for b1 being RealUnitarySpace
for b2 being Real
for b3 being sequence of b1 holds
( b3 is convergent implies lim (b2 * b3) = b2 * (lim b3) )
proof end;

theorem Th16: :: BHSP_2:16
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is convergent implies lim (- b2) = - (lim b2) )
proof end;

theorem Th17: :: BHSP_2:17
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent implies lim (b3 + b2) = (lim b3) + b2 )
proof end;

theorem Th18: :: BHSP_2:18
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent implies lim (b3 - b2) = (lim b3) - b2 )
proof end;

theorem Th19: :: BHSP_2:19
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent implies ( lim b3 = b2 iff for b4 being Real holds
not ( b4 > 0 & ( for b5 being Nat holds
ex b6 being Nat st
( b6 >= b5 & not ||.((b3 . b6) - b2).|| < b4 ) ) ) ) )
proof end;

definition
let c1 be RealUnitarySpace;
let c2 be sequence of c1;
func ||.c2.|| -> Real_Sequence means :Def3: :: BHSP_2:def 3
for b1 being Nat holds a3 . b1 = ||.(a2 . b1).||;
existence
ex b1 being Real_Sequence st
for b2 being Nat holds b1 . b2 = ||.(c2 . b2).||
proof end;
uniqueness
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds b1 . b3 = ||.(c2 . b3).|| ) & ( for b3 being Nat holds b2 . b3 = ||.(c2 . b3).|| ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines ||. BHSP_2:def 3 :
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Real_Sequence holds
( b3 = ||.b2.|| iff for b4 being Nat holds b3 . b4 = ||.(b2 . b4).|| );

theorem Th20: :: BHSP_2:20
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is convergent implies ||.b2.|| is convergent )
proof end;

theorem Th21: :: BHSP_2:21
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent & lim b3 = b2 implies ( ||.b3.|| is convergent & lim ||.b3.|| = ||.b2.|| ) )
proof end;

theorem Th22: :: BHSP_2:22
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent & lim b3 = b2 implies ( ||.(b3 - b2).|| is convergent & lim ||.(b3 - b2).|| = 0 ) )
proof end;

definition
let c1 be RealUnitarySpace;
let c2 be sequence of c1;
let c3 be Point of c1;
func dist c2,c3 -> Real_Sequence means :Def4: :: BHSP_2:def 4
for b1 being Nat holds a4 . b1 = dist (a2 . b1),a3;
existence
ex b1 being Real_Sequence st
for b2 being Nat holds b1 . b2 = dist (c2 . b2),c3
proof end;
uniqueness
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds b1 . b3 = dist (c2 . b3),c3 ) & ( for b3 being Nat holds b2 . b3 = dist (c2 . b3),c3 ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def4 defines dist BHSP_2:def 4 :
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Point of b1
for b4 being Real_Sequence holds
( b4 = dist b2,b3 iff for b5 being Nat holds b4 . b5 = dist (b2 . b5),b3 );

theorem Th23: :: BHSP_2:23
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent & lim b3 = b2 implies dist b3,b2 is convergent )
proof end;

theorem Th24: :: BHSP_2:24
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent & lim b3 = b2 implies ( dist b3,b2 is convergent & lim (dist b3,b2) = 0 ) )
proof end;

theorem Th25: :: BHSP_2:25
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4, b5 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 & b5 is convergent & lim b5 = b3 implies ( ||.(b4 + b5).|| is convergent & lim ||.(b4 + b5).|| = ||.(b2 + b3).|| ) )
proof end;

theorem Th26: :: BHSP_2:26
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4, b5 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 & b5 is convergent & lim b5 = b3 implies ( ||.((b4 + b5) - (b2 + b3)).|| is convergent & lim ||.((b4 + b5) - (b2 + b3)).|| = 0 ) )
proof end;

theorem Th27: :: BHSP_2:27
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4, b5 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 & b5 is convergent & lim b5 = b3 implies ( ||.(b4 - b5).|| is convergent & lim ||.(b4 - b5).|| = ||.(b2 - b3).|| ) )
proof end;

theorem Th28: :: BHSP_2:28
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4, b5 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 & b5 is convergent & lim b5 = b3 implies ( ||.((b4 - b5) - (b2 - b3)).|| is convergent & lim ||.((b4 - b5) - (b2 - b3)).|| = 0 ) )
proof end;

theorem Th29: :: BHSP_2:29
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real
for b4 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 implies ( ||.(b3 * b4).|| is convergent & lim ||.(b3 * b4).|| = ||.(b3 * b2).|| ) )
proof end;

theorem Th30: :: BHSP_2:30
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real
for b4 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 implies ( ||.((b3 * b4) - (b3 * b2)).|| is convergent & lim ||.((b3 * b4) - (b3 * b2)).|| = 0 ) )
proof end;

theorem Th31: :: BHSP_2:31
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent & lim b3 = b2 implies ( ||.(- b3).|| is convergent & lim ||.(- b3).|| = ||.(- b2).|| ) )
proof end;

theorem Th32: :: BHSP_2:32
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is convergent & lim b3 = b2 implies ( ||.((- b3) - (- b2)).|| is convergent & lim ||.((- b3) - (- b2)).|| = 0 ) )
proof end;

Lemma26: for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 implies ( ||.(b4 + b3).|| is convergent & lim ||.(b4 + b3).|| = ||.(b2 + b3).|| ) )
proof end;

theorem Th33: :: BHSP_2:33
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 implies ( ||.((b4 + b3) - (b2 + b3)).|| is convergent & lim ||.((b4 + b3) - (b2 + b3)).|| = 0 ) )
proof end;

theorem Th34: :: BHSP_2:34
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 implies ( ||.(b4 - b3).|| is convergent & lim ||.(b4 - b3).|| = ||.(b2 - b3).|| ) )
proof end;

theorem Th35: :: BHSP_2:35
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 implies ( ||.((b4 - b3) - (b2 - b3)).|| is convergent & lim ||.((b4 - b3) - (b2 - b3)).|| = 0 ) )
proof end;

theorem Th36: :: BHSP_2:36
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4, b5 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 & b5 is convergent & lim b5 = b3 implies ( dist (b4 + b5),(b2 + b3) is convergent & lim (dist (b4 + b5),(b2 + b3)) = 0 ) )
proof end;

theorem Th37: :: BHSP_2:37
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4, b5 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 & b5 is convergent & lim b5 = b3 implies ( dist (b4 - b5),(b2 - b3) is convergent & lim (dist (b4 - b5),(b2 - b3)) = 0 ) )
proof end;

theorem Th38: :: BHSP_2:38
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real
for b4 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 implies ( dist (b3 * b4),(b3 * b2) is convergent & lim (dist (b3 * b4),(b3 * b2)) = 0 ) )
proof end;

theorem Th39: :: BHSP_2:39
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being sequence of b1 holds
( b4 is convergent & lim b4 = b2 implies ( dist (b4 + b3),(b2 + b3) is convergent & lim (dist (b4 + b3),(b2 + b3)) = 0 ) )
proof end;

definition
let c1 be RealUnitarySpace;
let c2 be Point of c1;
let c3 be Real;
func Ball c2,c3 -> Subset of a1 equals :: BHSP_2:def 5
{ b1 where B is Point of a1 : ||.(a2 - b1).|| < a3 } ;
coherence
{ b1 where B is Point of c1 : ||.(c2 - b1).|| < c3 } is Subset of c1
proof end;
func cl_Ball c2,c3 -> Subset of a1 equals :: BHSP_2:def 6
{ b1 where B is Point of a1 : ||.(a2 - b1).|| <= a3 } ;
coherence
{ b1 where B is Point of c1 : ||.(c2 - b1).|| <= c3 } is Subset of c1
proof end;
func Sphere c2,c3 -> Subset of a1 equals :: BHSP_2:def 7
{ b1 where B is Point of a1 : ||.(a2 - b1).|| = a3 } ;
coherence
{ b1 where B is Point of c1 : ||.(c2 - b1).|| = c3 } is Subset of c1
proof end;
end;

:: deftheorem Def5 defines Ball BHSP_2:def 5 :
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real holds Ball b2,b3 = { b4 where B is Point of b1 : ||.(b2 - b4).|| < b3 } ;

:: deftheorem Def6 defines cl_Ball BHSP_2:def 6 :
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real holds cl_Ball b2,b3 = { b4 where B is Point of b1 : ||.(b2 - b4).|| <= b3 } ;

:: deftheorem Def7 defines Sphere BHSP_2:def 7 :
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real holds Sphere b2,b3 = { b4 where B is Point of b1 : ||.(b2 - b4).|| = b3 } ;

theorem Th40: :: BHSP_2:40
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
( b2 in Ball b3,b4 iff ||.(b3 - b2).|| < b4 )
proof end;

theorem Th41: :: BHSP_2:41
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
( b2 in Ball b3,b4 iff dist b3,b2 < b4 )
proof end;

theorem Th42: :: BHSP_2:42
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real holds
( b3 > 0 implies b2 in Ball b2,b3 )
proof end;

theorem Th43: :: BHSP_2:43
for b1 being RealUnitarySpace
for b2, b3, b4 being Point of b1
for b5 being Real holds
not ( b2 in Ball b3,b5 & b4 in Ball b3,b5 & not dist b2,b4 < 2 * b5 )
proof end;

theorem Th44: :: BHSP_2:44
for b1 being RealUnitarySpace
for b2, b3, b4 being Point of b1
for b5 being Real holds
( b2 in Ball b3,b5 implies b2 - b4 in Ball (b3 - b4),b5 )
proof end;

theorem Th45: :: BHSP_2:45
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
( b2 in Ball b3,b4 implies b2 - b3 in Ball (0. b1),b4 )
proof end;

theorem Th46: :: BHSP_2:46
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4, b5 being Real holds
( b2 in Ball b3,b4 & b4 <= b5 implies b2 in Ball b3,b5 )
proof end;

theorem Th47: :: BHSP_2:47
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
( b2 in cl_Ball b3,b4 iff ||.(b3 - b2).|| <= b4 )
proof end;

theorem Th48: :: BHSP_2:48
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
( b2 in cl_Ball b3,b4 iff dist b3,b2 <= b4 )
proof end;

theorem Th49: :: BHSP_2:49
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real holds
( b3 >= 0 implies b2 in cl_Ball b2,b3 )
proof end;

theorem Th50: :: BHSP_2:50
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
( b2 in Ball b3,b4 implies b2 in cl_Ball b3,b4 )
proof end;

theorem Th51: :: BHSP_2:51
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
( b2 in Sphere b3,b4 iff ||.(b3 - b2).|| = b4 )
proof end;

theorem Th52: :: BHSP_2:52
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
( b2 in Sphere b3,b4 iff dist b3,b2 = b4 )
proof end;

theorem Th53: :: BHSP_2:53
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
( b2 in Sphere b3,b4 implies b2 in cl_Ball b3,b4 )
proof end;

theorem Th54: :: BHSP_2:54
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real holds Ball b2,b3 c= cl_Ball b2,b3
proof end;

theorem Th55: :: BHSP_2:55
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real holds Sphere b2,b3 c= cl_Ball b2,b3
proof end;

theorem Th56: :: BHSP_2:56
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real holds (Ball b2,b3) \/ (Sphere b2,b3) = cl_Ball b2,b3
proof end;