:: RUSUB_5 semantic presentation

definition
let c1 be non empty RLSStruct ;
let c2, c3 be Affine Subset of c1;
pred c2 is_parallel_to c3 means :Def1: :: RUSUB_5:def 1
ex b1 being VECTOR of a1 st a2 = a3 + {b1};
end;

:: deftheorem Def1 defines is_parallel_to RUSUB_5:def 1 :
for b1 being non empty RLSStruct
for b2, b3 being Affine Subset of b1 holds
( b2 is_parallel_to b3 iff ex b4 being VECTOR of b1 st b2 = b3 + {b4} );

theorem Th1: :: RUSUB_5:1
for b1 being non empty right_zeroed RLSStruct
for b2 being Affine Subset of b1 holds b2 is_parallel_to b2
proof end;

theorem Th2: :: RUSUB_5:2
for b1 being non empty add-associative right_zeroed right_complementable RLSStruct
for b2, b3 being Affine Subset of b1 holds
( b2 is_parallel_to b3 implies b3 is_parallel_to b2 )
proof end;

theorem Th3: :: RUSUB_5:3
for b1 being non empty Abelian add-associative right_zeroed right_complementable RLSStruct
for b2, b3, b4 being Affine Subset of b1 holds
( b2 is_parallel_to b3 & b3 is_parallel_to b4 implies b2 is_parallel_to b4 )
proof end;

definition
let c1 be non empty LoopStr ;
let c2, c3 be Subset of c1;
func c2 - c3 -> Subset of a1 equals :: RUSUB_5:def 2
{ (b1 - b2) where B is Element of a1, B is Element of a1 : ( b1 in a2 & b2 in a3 ) } ;
coherence
{ (b1 - b2) where B is Element of c1, B is Element of c1 : ( b1 in c2 & b2 in c3 ) } is Subset of c1
proof end;
end;

:: deftheorem Def2 defines - RUSUB_5:def 2 :
for b1 being non empty LoopStr
for b2, b3 being Subset of b1 holds b2 - b3 = { (b4 - b5) where B is Element of b1, B is Element of b1 : ( b4 in b2 & b5 in b3 ) } ;

theorem Th4: :: RUSUB_5:4
for b1 being RealLinearSpace
for b2, b3 being Affine Subset of b1 holds b2 - b3 is Affine
proof end;

theorem Th5: :: RUSUB_5:5
for b1 being non empty LoopStr
for b2, b3 being Subset of b1 holds
( ( b2 is empty or b3 is empty ) implies b2 + b3 is empty )
proof end;

theorem Th6: :: RUSUB_5:6
for b1 being non empty LoopStr
for b2, b3 being non empty Subset of b1 holds
not b2 + b3 is empty
proof end;

theorem Th7: :: RUSUB_5:7
for b1 being non empty LoopStr
for b2, b3 being Subset of b1 holds
( ( b2 is empty or b3 is empty ) implies b2 - b3 is empty )
proof end;

theorem Th8: :: RUSUB_5:8
for b1 being non empty LoopStr
for b2, b3 being non empty Subset of b1 holds
not b2 - b3 is empty
proof end;

theorem Th9: :: RUSUB_5:9
for b1 being non empty Abelian add-associative right_zeroed right_complementable LoopStr
for b2, b3 being Subset of b1
for b4 being Element of b1 holds
( b2 = b3 + {b4} iff b2 - {b4} = b3 )
proof end;

theorem Th10: :: RUSUB_5:10
for b1 being non empty Abelian add-associative right_zeroed right_complementable LoopStr
for b2, b3 being Subset of b1
for b4 being Element of b1 holds
( b4 in b3 implies b2 + {b4} c= b2 + b3 )
proof end;

theorem Th11: :: RUSUB_5:11
for b1 being non empty Abelian add-associative right_zeroed right_complementable LoopStr
for b2, b3 being Subset of b1
for b4 being Element of b1 holds
( b4 in b3 implies b2 - {b4} c= b2 - b3 )
proof end;

theorem Th12: :: RUSUB_5:12
for b1 being RealLinearSpace
for b2 being non empty Subset of b1 holds 0. b1 in b2 - b2
proof end;

theorem Th13: :: RUSUB_5:13
for b1 being RealLinearSpace
for b2 being non empty Affine Subset of b1
for b3 being VECTOR of b1 holds
( b2 is Subspace-like & b3 in b2 implies b2 + {b3} c= b2 )
proof end;

theorem Th14: :: RUSUB_5:14
for b1 being RealLinearSpace
for b2, b3, b4 being non empty Affine Subset of b1 holds
( b3 is Subspace-like & b4 is Subspace-like & b2 is_parallel_to b3 & b2 is_parallel_to b4 implies b3 = b4 )
proof end;

theorem Th15: :: RUSUB_5:15
for b1 being RealLinearSpace
for b2 being non empty Affine Subset of b1
for b3 being VECTOR of b1 holds
( b3 in b2 implies 0. b1 in b2 - {b3} )
proof end;

theorem Th16: :: RUSUB_5:16
for b1 being RealLinearSpace
for b2 being non empty Affine Subset of b1
for b3 being VECTOR of b1 holds
not ( b3 in b2 & ( for b4 being non empty Affine Subset of b1 holds
not ( b4 = b2 - {b3} & b2 is_parallel_to b4 & b4 is Subspace-like ) ) )
proof end;

theorem Th17: :: RUSUB_5:17
for b1 being RealLinearSpace
for b2 being non empty Affine Subset of b1
for b3, b4 being VECTOR of b1 holds
( b3 in b2 & b4 in b2 implies b2 - {b4} = b2 - {b3} )
proof end;

theorem Th18: :: RUSUB_5:18
for b1 being RealLinearSpace
for b2 being non empty Affine Subset of b1 holds b2 - b2 = union { (b2 - {b3}) where B is VECTOR of b1 : b3 in b2 }
proof end;

theorem Th19: :: RUSUB_5:19
for b1 being RealLinearSpace
for b2 being non empty Affine Subset of b1
for b3 being VECTOR of b1 holds
( b3 in b2 implies b2 - {b3} = union { (b2 - {b4}) where B is VECTOR of b1 : b4 in b2 } )
proof end;

theorem Th20: :: RUSUB_5:20
for b1 being RealLinearSpace
for b2 being non empty Affine Subset of b1 holds
ex b3 being non empty Affine Subset of b1 st
( b3 = b2 - b2 & b3 is Subspace-like & b2 is_parallel_to b3 )
proof end;

definition
let c1 be RealUnitarySpace;
let c2 be Subspace of c1;
func Ort_Comp c2 -> strict Subspace of a1 means :Def3: :: RUSUB_5:def 3
the carrier of a3 = { b1 where B is VECTOR of a1 : for b1 being VECTOR of a1 holds
( b2 in a2 implies b2,b1 are_orthogonal )
}
;
existence
ex b1 being strict Subspace of c1 st the carrier of b1 = { b2 where B is VECTOR of c1 : for b1 being VECTOR of c1 holds
( b3 in c2 implies b3,b2 are_orthogonal )
}
proof end;
uniqueness
for b1, b2 being strict Subspace of c1 holds
( the carrier of b1 = { b3 where B is VECTOR of c1 : for b1 being VECTOR of c1 holds
( b4 in c2 implies b4,b3 are_orthogonal )
}
& the carrier of b2 = { b3 where B is VECTOR of c1 : for b1 being VECTOR of c1 holds
( b4 in c2 implies b4,b3 are_orthogonal )
}
implies b1 = b2 )
by RUSUB_1:24;
end;

:: deftheorem Def3 defines Ort_Comp RUSUB_5:def 3 :
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being strict Subspace of b1 holds
( b3 = Ort_Comp b2 iff the carrier of b3 = { b4 where B is VECTOR of b1 : for b1 being VECTOR of b1 holds
( b5 in b2 implies b5,b4 are_orthogonal )
}
);

definition
let c1 be RealUnitarySpace;
let c2 be non empty Subset of c1;
func Ort_Comp c2 -> strict Subspace of a1 means :Def4: :: RUSUB_5:def 4
the carrier of a3 = { b1 where B is VECTOR of a1 : for b1 being VECTOR of a1 holds
( b2 in a2 implies b2,b1 are_orthogonal )
}
;
existence
ex b1 being strict Subspace of c1 st the carrier of b1 = { b2 where B is VECTOR of c1 : for b1 being VECTOR of c1 holds
( b3 in c2 implies b3,b2 are_orthogonal )
}
proof end;
uniqueness
for b1, b2 being strict Subspace of c1 holds
( the carrier of b1 = { b3 where B is VECTOR of c1 : for b1 being VECTOR of c1 holds
( b4 in c2 implies b4,b3 are_orthogonal )
}
& the carrier of b2 = { b3 where B is VECTOR of c1 : for b1 being VECTOR of c1 holds
( b4 in c2 implies b4,b3 are_orthogonal )
}
implies b1 = b2 )
by RUSUB_1:24;
end;

:: deftheorem Def4 defines Ort_Comp RUSUB_5:def 4 :
for b1 being RealUnitarySpace
for b2 being non empty Subset of b1
for b3 being strict Subspace of b1 holds
( b3 = Ort_Comp b2 iff the carrier of b3 = { b4 where B is VECTOR of b1 : for b1 being VECTOR of b1 holds
( b5 in b2 implies b5,b4 are_orthogonal )
}
);

theorem Th21: :: RUSUB_5:21
for b1 being RealUnitarySpace
for b2 being Subspace of b1 holds 0. b1 in Ort_Comp b2
proof end;

theorem Th22: :: RUSUB_5:22
for b1 being RealUnitarySpace holds Ort_Comp ((0). b1) = (Omega). b1
proof end;

theorem Th23: :: RUSUB_5:23
for b1 being RealUnitarySpace holds Ort_Comp ((Omega). b1) = (0). b1
proof end;

theorem Th24: :: RUSUB_5:24
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being VECTOR of b1 holds
not ( b3 <> 0. b1 & b3 in b2 & b3 in Ort_Comp b2 )
proof end;

theorem Th25: :: RUSUB_5:25
for b1 being RealUnitarySpace
for b2 being non empty Subset of b1 holds b2 c= the carrier of (Ort_Comp (Ort_Comp b2))
proof end;

theorem Th26: :: RUSUB_5:26
for b1 being RealUnitarySpace
for b2, b3 being non empty Subset of b1 holds
( b2 c= b3 implies the carrier of (Ort_Comp b3) c= the carrier of (Ort_Comp b2) )
proof end;

theorem Th27: :: RUSUB_5:27
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being non empty Subset of b1 holds
( b3 = the carrier of b2 implies Ort_Comp b3 = Ort_Comp b2 )
proof end;

theorem Th28: :: RUSUB_5:28
for b1 being RealUnitarySpace
for b2 being non empty Subset of b1 holds Ort_Comp b2 = Ort_Comp (Ort_Comp (Ort_Comp b2))
proof end;

theorem Th29: :: RUSUB_5:29
for b1 being RealUnitarySpace
for b2, b3 being VECTOR of b1 holds
( ||.(b2 + b3).|| ^2 = ((||.b2.|| ^2 ) + (2 * (b2 .|. b3))) + (||.b3.|| ^2 ) & ||.(b2 - b3).|| ^2 = ((||.b2.|| ^2 ) - (2 * (b2 .|. b3))) + (||.b3.|| ^2 ) )
proof end;

theorem Th30: :: RUSUB_5:30
for b1 being RealUnitarySpace
for b2, b3 being VECTOR of b1 holds
( b2,b3 are_orthogonal implies ||.(b2 + b3).|| ^2 = (||.b2.|| ^2 ) + (||.b3.|| ^2 ) )
proof end;

theorem Th31: :: RUSUB_5:31
for b1 being RealUnitarySpace
for b2, b3 being VECTOR of b1 holds (||.(b2 + b3).|| ^2 ) + (||.(b2 - b3).|| ^2 ) = (2 * (||.b2.|| ^2 )) + (2 * (||.b3.|| ^2 ))
proof end;

theorem Th32: :: RUSUB_5:32
for b1 being RealUnitarySpace
for b2 being VECTOR of b1 holds
ex b3 being Subspace of b1 st the carrier of b3 = { b4 where B is VECTOR of b1 : b4 .|. b2 = 0 }
proof end;

definition
let c1 be RealUnitarySpace;
func Family_open_set c1 -> Subset-Family of a1 means :Def5: :: RUSUB_5:def 5
for b1 being Subset of a1 holds
( b1 in a2 iff for b2 being Point of a1 holds
not ( b2 in b1 & ( for b3 being Real holds
not ( b3 > 0 & Ball b2,b3 c= b1 ) ) ) );
existence
ex b1 being Subset-Family of c1 st
for b2 being Subset of c1 holds
( b2 in b1 iff for b3 being Point of c1 holds
not ( b3 in b2 & ( for b4 being Real holds
not ( b4 > 0 & Ball b3,b4 c= b2 ) ) ) )
proof end;
uniqueness
for b1, b2 being Subset-Family of c1 holds
( ( for b3 being Subset of c1 holds
( b3 in b1 iff for b4 being Point of c1 holds
not ( b4 in b3 & ( for b5 being Real holds
not ( b5 > 0 & Ball b4,b5 c= b3 ) ) ) ) ) & ( for b3 being Subset of c1 holds
( b3 in b2 iff for b4 being Point of c1 holds
not ( b4 in b3 & ( for b5 being Real holds
not ( b5 > 0 & Ball b4,b5 c= b3 ) ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def5 defines Family_open_set RUSUB_5:def 5 :
for b1 being RealUnitarySpace
for b2 being Subset-Family of b1 holds
( b2 = Family_open_set b1 iff for b3 being Subset of b1 holds
( b3 in b2 iff for b4 being Point of b1 holds
not ( b4 in b3 & ( for b5 being Real holds
not ( b5 > 0 & Ball b4,b5 c= b3 ) ) ) ) );

theorem Th33: :: RUSUB_5:33
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3, b4 being Real holds
( b3 <= b4 implies Ball b2,b3 c= Ball b2,b4 )
proof end;

theorem Th34: :: RUSUB_5:34
for b1 being RealUnitarySpace
for b2 being Point of b1 holds
ex b3 being Real st
( b3 > 0 & Ball b2,b3 c= the carrier of b1 )
proof end;

theorem Th35: :: RUSUB_5:35
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4 being Real holds
not ( b3 in Ball b2,b4 & ( for b5 being Real holds
not ( b5 > 0 & Ball b3,b5 c= Ball b2,b4 ) ) )
proof end;

theorem Th36: :: RUSUB_5:36
for b1 being RealUnitarySpace
for b2, b3, b4 being Point of b1
for b5, b6 being Real holds
not ( b3 in (Ball b2,b5) /\ (Ball b4,b6) & ( for b7 being Real holds
not ( Ball b3,b7 c= Ball b2,b5 & Ball b3,b7 c= Ball b4,b6 ) ) )
proof end;

theorem Th37: :: RUSUB_5:37
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being Real holds Ball b2,b3 in Family_open_set b1
proof end;

theorem Th38: :: RUSUB_5:38
for b1 being RealUnitarySpace holds the carrier of b1 in Family_open_set b1
proof end;

theorem Th39: :: RUSUB_5:39
for b1 being RealUnitarySpace
for b2, b3 being Subset of b1 holds
( b2 in Family_open_set b1 & b3 in Family_open_set b1 implies b2 /\ b3 in Family_open_set b1 )
proof end;

theorem Th40: :: RUSUB_5:40
for b1 being RealUnitarySpace
for b2 being Subset-Family of b1 holds
( b2 c= Family_open_set b1 implies union b2 in Family_open_set b1 )
proof end;

theorem Th41: :: RUSUB_5:41
for b1 being RealUnitarySpace holds
TopStruct(# the carrier of b1,(Family_open_set b1) #) is TopSpace
proof end;

definition
let c1 be RealUnitarySpace;
func TopUnitSpace c1 -> TopStruct equals :: RUSUB_5:def 6
TopStruct(# the carrier of a1,(Family_open_set a1) #);
coherence
TopStruct(# the carrier of c1,(Family_open_set c1) #) is TopStruct
;
end;

:: deftheorem Def6 defines TopUnitSpace RUSUB_5:def 6 :
for b1 being RealUnitarySpace holds TopUnitSpace b1 = TopStruct(# the carrier of b1,(Family_open_set b1) #);

registration
let c1 be RealUnitarySpace;
cluster TopUnitSpace a1 -> TopSpace-like ;
coherence
TopUnitSpace c1 is TopSpace-like
by Th41;
end;

registration
let c1 be RealUnitarySpace;
cluster TopUnitSpace a1 -> non empty TopSpace-like ;
coherence
not TopUnitSpace c1 is empty
;
end;

theorem Th42: :: RUSUB_5:42
for b1 being RealUnitarySpace
for b2 being Subset of (TopUnitSpace b1) holds
( b2 = [#] b1 implies ( b2 is open & b2 is closed ) )
proof end;

theorem Th43: :: RUSUB_5:43
for b1 being RealUnitarySpace
for b2 being Subset of (TopUnitSpace b1) holds
( b2 = {} b1 implies ( b2 is open & b2 is closed ) )
proof end;

theorem Th44: :: RUSUB_5:44
for b1 being RealUnitarySpace
for b2 being VECTOR of b1
for b3 being Real holds
( the carrier of b1 = {(0. b1)} & b3 <> 0 implies Sphere b2,b3 is empty )
proof end;

theorem Th45: :: RUSUB_5:45
for b1 being RealUnitarySpace
for b2 being VECTOR of b1
for b3 being Real holds
not ( the carrier of b1 <> {(0. b1)} & b3 > 0 & Sphere b2,b3 is empty )
proof end;

theorem Th46: :: RUSUB_5:46
for b1 being RealUnitarySpace
for b2 being VECTOR of b1
for b3 being Real holds
( b3 = 0 implies Ball b2,b3 is empty )
proof end;

theorem Th47: :: RUSUB_5:47
for b1 being RealUnitarySpace
for b2 being VECTOR of b1
for b3 being Real holds
( the carrier of b1 = {(0. b1)} & b3 > 0 implies Ball b2,b3 = {(0. b1)} )
proof end;

theorem Th48: :: RUSUB_5:48
for b1 being RealUnitarySpace
for b2 being VECTOR of b1
for b3 being Real holds
not ( the carrier of b1 <> {(0. b1)} & b3 > 0 & ( for b4 being VECTOR of b1 holds
not ( b4 <> b2 & b4 in Ball b2,b3 ) ) )
proof end;

theorem Th49: :: RUSUB_5:49
for b1 being RealUnitarySpace holds
( the carrier of b1 = the carrier of (TopUnitSpace b1) & the topology of (TopUnitSpace b1) = Family_open_set b1 ) ;

theorem Th50: :: RUSUB_5:50
for b1 being RealUnitarySpace
for b2 being Subset of (TopUnitSpace b1)
for b3 being Real
for b4 being Point of b1 holds
( b2 = Ball b4,b3 implies b2 is open )
proof end;

theorem Th51: :: RUSUB_5:51
for b1 being RealUnitarySpace
for b2 being Subset of (TopUnitSpace b1) holds
( b2 is open iff for b3 being Point of b1 holds
not ( b3 in b2 & ( for b4 being Real holds
not ( b4 > 0 & Ball b3,b4 c= b2 ) ) ) )
proof end;

theorem Th52: :: RUSUB_5:52
for b1 being RealUnitarySpace
for b2, b3 being Point of b1
for b4, b5 being Real holds
ex b6 being Point of b1ex b7 being Real st (Ball b2,b4) \/ (Ball b3,b5) c= Ball b6,b7
proof end;

theorem Th53: :: RUSUB_5:53
for b1 being RealUnitarySpace
for b2 being Subset of (TopUnitSpace b1)
for b3 being VECTOR of b1
for b4 being Real holds
( b2 = cl_Ball b3,b4 implies b2 is closed )
proof end;

theorem Th54: :: RUSUB_5:54
for b1 being RealUnitarySpace
for b2 being Subset of (TopUnitSpace b1)
for b3 being VECTOR of b1
for b4 being Real holds
( b2 = Sphere b3,b4 implies b2 is closed )
proof end;