:: BHSP_6 semantic presentation

definition
let c1 be RealUnitarySpace;
assume E1: ( the add of c1 is commutative & the add of c1 is associative & the add of c1 has_a_unity ) ;
let c2 be finite Subset of c1;
func setsum c2 -> Element of a1 means :Def1: :: BHSP_6:def 1
ex b1 being FinSequence of the carrier of a1 st
( b1 is one-to-one & rng b1 = a2 & a3 = the add of a1 "**" b1 );
existence
ex b1 being Element of c1ex b2 being FinSequence of the carrier of c1 st
( b2 is one-to-one & rng b2 = c2 & b1 = the add of c1 "**" b2 )
proof end;
uniqueness
for b1, b2 being Element of c1 holds
( ex b3 being FinSequence of the carrier of c1 st
( b3 is one-to-one & rng b3 = c2 & b1 = the add of c1 "**" b3 ) & ex b3 being FinSequence of the carrier of c1 st
( b3 is one-to-one & rng b3 = c2 & b2 = the add of c1 "**" b3 ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def1 defines setsum BHSP_6:def 1 :
for b1 being RealUnitarySpace holds
( the add of b1 is commutative & the add of b1 is associative & the add of b1 has_a_unity implies for b2 being finite Subset of b1
for b3 being Element of b1 holds
( b3 = setsum b2 iff ex b4 being FinSequence of the carrier of b1 st
( b4 is one-to-one & rng b4 = b2 & b3 = the add of b1 "**" b4 ) ) );

theorem Th1: :: BHSP_6:1
for b1 being RealUnitarySpace holds
( the add of b1 is commutative & the add of b1 is associative & the add of b1 has_a_unity implies for b2 being finite Subset of b1
for b3 being Function of the carrier of b1,the carrier of b1 holds
( b2 c= dom b3 & ( for b4 being set holds
( b4 in dom b3 implies b3 . b4 = b4 ) ) implies setsum b2 = setopfunc b2,the carrier of b1,the carrier of b1,b3,the add of b1 ) )
proof end;

theorem Th2: :: BHSP_6:2
for b1 being RealUnitarySpace holds
( the add of b1 is commutative & the add of b1 is associative & the add of b1 has_a_unity implies for b2, b3 being finite Subset of b1 holds
( b2 misses b3 implies for b4 being finite Subset of b1 holds
( b4 = b2 \/ b3 implies setsum b4 = (setsum b2) + (setsum b3) ) ) )
proof end;

definition
let c1 be RealUnitarySpace;
let c2 be Subset of c1;
attr a2 is summable_set means :Def2: :: BHSP_6:def 2
ex b1 being Point of a1 st
for b2 being Real holds
not ( b2 > 0 & ( for b3 being finite Subset of a1 holds
not ( not b3 is empty & b3 c= a2 & ( for b4 being finite Subset of a1 holds
not ( b3 c= b4 & b4 c= a2 & not ||.(b1 - (setsum b4)).|| < b2 ) ) ) ) );
end;

:: deftheorem Def2 defines summable_set BHSP_6:def 2 :
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
( b2 is summable_set iff ex b3 being Point of b1 st
for b4 being Real holds
not ( b4 > 0 & ( for b5 being finite Subset of b1 holds
not ( not b5 is empty & b5 c= b2 & ( for b6 being finite Subset of b1 holds
not ( b5 c= b6 & b6 c= b2 & not ||.(b3 - (setsum b6)).|| < b4 ) ) ) ) ) );

definition
let c1 be RealUnitarySpace;
let c2 be Subset of c1;
assume E5: c2 is summable_set ;
func sum c2 -> Point of a1 means :: BHSP_6:def 3
for b1 being Real holds
not ( b1 > 0 & ( for b2 being finite Subset of a1 holds
not ( not b2 is empty & b2 c= a2 & ( for b3 being finite Subset of a1 holds
not ( b2 c= b3 & b3 c= a2 & not ||.(a3 - (setsum b3)).|| < b1 ) ) ) ) );
existence
ex b1 being Point of c1 st
for b2 being Real holds
not ( b2 > 0 & ( for b3 being finite Subset of c1 holds
not ( not b3 is empty & b3 c= c2 & ( for b4 being finite Subset of c1 holds
not ( b3 c= b4 & b4 c= c2 & not ||.(b1 - (setsum b4)).|| < b2 ) ) ) ) )
by E5, Def2;
uniqueness
for b1, b2 being Point of c1 holds
( ( for b3 being Real holds
not ( b3 > 0 & ( for b4 being finite Subset of c1 holds
not ( not b4 is empty & b4 c= c2 & ( for b5 being finite Subset of c1 holds
not ( b4 c= b5 & b5 c= c2 & not ||.(b1 - (setsum b5)).|| < b3 ) ) ) ) ) ) & ( for b3 being Real holds
not ( b3 > 0 & ( for b4 being finite Subset of c1 holds
not ( not b4 is empty & b4 c= c2 & ( for b5 being finite Subset of c1 holds
not ( b4 c= b5 & b5 c= c2 & not ||.(b2 - (setsum b5)).|| < b3 ) ) ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines sum BHSP_6:def 3 :
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
( b2 is summable_set implies for b3 being Point of b1 holds
( b3 = sum b2 iff for b4 being Real holds
not ( b4 > 0 & ( for b5 being finite Subset of b1 holds
not ( not b5 is empty & b5 c= b2 & ( for b6 being finite Subset of b1 holds
not ( b5 c= b6 & b6 c= b2 & not ||.(b3 - (setsum b6)).|| < b4 ) ) ) ) ) ) );

definition
let c1 be RealUnitarySpace;
let c2 be linear-Functional of c1;
attr a2 is Bounded means :Def4: :: BHSP_6:def 4
ex b1 being Real st
( b1 > 0 & ( for b2 being Point of a1 holds abs (a2 . b2) <= b1 * ||.b2.|| ) );
end;

:: deftheorem Def4 defines Bounded BHSP_6:def 4 :
for b1 being RealUnitarySpace
for b2 being linear-Functional of b1 holds
( b2 is Bounded iff ex b3 being Real st
( b3 > 0 & ( for b4 being Point of b1 holds abs (b2 . b4) <= b3 * ||.b4.|| ) ) );

definition
let c1 be RealUnitarySpace;
let c2 be Subset of c1;
attr a2 is weakly_summable_set means :Def5: :: BHSP_6:def 5
ex b1 being Point of a1 st
for b2 being linear-Functional of a1 holds
( b2 is Bounded implies for b3 being Real holds
not ( b3 > 0 & ( for b4 being finite Subset of a1 holds
not ( not b4 is empty & b4 c= a2 & ( for b5 being finite Subset of a1 holds
not ( b4 c= b5 & b5 c= a2 & not abs (b2 . (b1 - (setsum b5))) < b3 ) ) ) ) ) );
end;

:: deftheorem Def5 defines weakly_summable_set BHSP_6:def 5 :
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
( b2 is weakly_summable_set iff ex b3 being Point of b1 st
for b4 being linear-Functional of b1 holds
( b4 is Bounded implies for b5 being Real holds
not ( b5 > 0 & ( for b6 being finite Subset of b1 holds
not ( not b6 is empty & b6 c= b2 & ( for b7 being finite Subset of b1 holds
not ( b6 c= b7 & b7 c= b2 & not abs (b4 . (b3 - (setsum b7))) < b5 ) ) ) ) ) ) );

definition
let c1 be RealUnitarySpace;
let c2 be Subset of c1;
let c3 be Functional of c1;
pred c2 is_summable_set_by c3 means :Def6: :: BHSP_6:def 6
ex b1 being Real st
for b2 being Real holds
not ( b2 > 0 & ( for b3 being finite Subset of a1 holds
not ( not b3 is empty & b3 c= a2 & ( for b4 being finite Subset of a1 holds
not ( b3 c= b4 & b4 c= a2 & not abs (b1 - (setopfunc b4,the carrier of a1,REAL ,a3,addreal )) < b2 ) ) ) ) );
end;

:: deftheorem Def6 defines is_summable_set_by BHSP_6:def 6 :
for b1 being RealUnitarySpace
for b2 being Subset of b1
for b3 being Functional of b1 holds
( b2 is_summable_set_by b3 iff ex b4 being Real st
for b5 being Real holds
not ( b5 > 0 & ( for b6 being finite Subset of b1 holds
not ( not b6 is empty & b6 c= b2 & ( for b7 being finite Subset of b1 holds
not ( b6 c= b7 & b7 c= b2 & not abs (b4 - (setopfunc b7,the carrier of b1,REAL ,b3,addreal )) < b5 ) ) ) ) ) );

definition
let c1 be RealUnitarySpace;
let c2 be Subset of c1;
let c3 be Functional of c1;
assume E8: c2 is_summable_set_by c3 ;
func sum_byfunc c2,c3 -> Real means :: BHSP_6:def 7
for b1 being Real holds
not ( b1 > 0 & ( for b2 being finite Subset of a1 holds
not ( not b2 is empty & b2 c= a2 & ( for b3 being finite Subset of a1 holds
not ( b2 c= b3 & b3 c= a2 & not abs (a4 - (setopfunc b3,the carrier of a1,REAL ,a3,addreal )) < b1 ) ) ) ) );
existence
ex b1 being Real st
for b2 being Real holds
not ( b2 > 0 & ( for b3 being finite Subset of c1 holds
not ( not b3 is empty & b3 c= c2 & ( for b4 being finite Subset of c1 holds
not ( b3 c= b4 & b4 c= c2 & not abs (b1 - (setopfunc b4,the carrier of c1,REAL ,c3,addreal )) < b2 ) ) ) ) )
by E8, Def6;
uniqueness
for b1, b2 being Real holds
( ( for b3 being Real holds
not ( b3 > 0 & ( for b4 being finite Subset of c1 holds
not ( not b4 is empty & b4 c= c2 & ( for b5 being finite Subset of c1 holds
not ( b4 c= b5 & b5 c= c2 & not abs (b1 - (setopfunc b5,the carrier of c1,REAL ,c3,addreal )) < b3 ) ) ) ) ) ) & ( for b3 being Real holds
not ( b3 > 0 & ( for b4 being finite Subset of c1 holds
not ( not b4 is empty & b4 c= c2 & ( for b5 being finite Subset of c1 holds
not ( b4 c= b5 & b5 c= c2 & not abs (b2 - (setopfunc b5,the carrier of c1,REAL ,c3,addreal )) < b3 ) ) ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def7 defines sum_byfunc BHSP_6:def 7 :
for b1 being RealUnitarySpace
for b2 being Subset of b1
for b3 being Functional of b1 holds
( b2 is_summable_set_by b3 implies for b4 being Real holds
( b4 = sum_byfunc b2,b3 iff for b5 being Real holds
not ( b5 > 0 & ( for b6 being finite Subset of b1 holds
not ( not b6 is empty & b6 c= b2 & ( for b7 being finite Subset of b1 holds
not ( b6 c= b7 & b7 c= b2 & not abs (b4 - (setopfunc b7,the carrier of b1,REAL ,b3,addreal )) < b5 ) ) ) ) ) ) );

theorem Th3: :: BHSP_6:3
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
( b2 is summable_set implies b2 is weakly_summable_set )
proof end;

theorem Th4: :: BHSP_6:4
for b1 being RealUnitarySpace
for b2 being linear-Functional of b1
for b3 being FinSequence of the carrier of b1 holds
( len b3 >= 1 implies for b4 being FinSequence of REAL holds
( dom b3 = dom b4 & ( for b5 being Nat holds
( b5 in dom b4 implies b4 . b5 = b2 . (b3 . b5) ) ) implies b2 . (the add of b1 "**" b3) = addreal "**" b4 ) )
proof end;

theorem Th5: :: BHSP_6:5
for b1 being RealUnitarySpace holds
( the add of b1 is commutative & the add of b1 is associative & the add of b1 has_a_unity implies for b2 being finite Subset of b1 holds
( not b2 is empty implies for b3 being linear-Functional of b1 holds b3 . (setsum b2) = setopfunc b2,the carrier of b1,REAL ,b3,addreal ) )
proof end;

theorem Th6: :: BHSP_6:6
for b1 being RealUnitarySpace holds
( the add of b1 is commutative & the add of b1 is associative & the add of b1 has_a_unity implies for b2 being Subset of b1 holds
not ( b2 is weakly_summable_set & ( for b3 being Point of b1 holds
ex b4 being linear-Functional of b1 st
( b4 is Bounded & ex b5 being Real st
( b5 > 0 & ( for b6 being finite Subset of b1 holds
not ( not b6 is empty & b6 c= b2 & ( for b7 being finite Subset of b1 holds
not ( b6 c= b7 & b7 c= b2 & not abs ((b4 . b3) - (setopfunc b7,the carrier of b1,REAL ,b4,addreal )) < b5 ) ) ) ) ) ) ) ) )
proof end;

theorem Th7: :: BHSP_6:7
for b1 being RealUnitarySpace holds
( the add of b1 is commutative & the add of b1 is associative & the add of b1 has_a_unity implies for b2 being Subset of b1 holds
( b2 is weakly_summable_set implies for b3 being linear-Functional of b1 holds
( b3 is Bounded implies b2 is_summable_set_by b3 ) ) )
proof end;

theorem Th8: :: BHSP_6:8
for b1 being RealUnitarySpace holds
( the add of b1 is commutative & the add of b1 is associative & the add of b1 has_a_unity implies for b2 being Subset of b1 holds
( b2 is summable_set implies for b3 being linear-Functional of b1 holds
( b3 is Bounded implies b2 is_summable_set_by b3 ) ) )
proof end;

theorem Th9: :: BHSP_6:9
for b1 being RealUnitarySpace
for b2 being finite Subset of b1 holds
( not b2 is empty implies b2 is summable_set )
proof end;

theorem Th10: :: BHSP_6:10
for b1 being RealUnitarySpace holds
( the add of b1 is commutative & the add of b1 is associative & the add of b1 has_a_unity & b1 is_Hilbert_space implies for b2 being Subset of b1 holds
( b2 is summable_set iff for b3 being Real holds
not ( b3 > 0 & ( for b4 being finite Subset of b1 holds
not ( not b4 is empty & b4 c= b2 & ( for b5 being finite Subset of b1 holds
not ( not b5 is empty & b5 c= b2 & b4 misses b5 & not ||.(setsum b5).|| < b3 ) ) ) ) ) ) )
proof end;