:: BHSP_6 semantic presentation
:: deftheorem Def1 defines setsum BHSP_6:def 1 :
theorem Th1: :: BHSP_6:1
theorem Th2: :: BHSP_6:2
:: deftheorem Def2 defines summable_set BHSP_6:def 2 :
:: deftheorem Def3 defines sum BHSP_6:def 3 :
:: deftheorem Def4 defines Bounded BHSP_6:def 4 :
:: deftheorem Def5 defines weakly_summable_set BHSP_6:def 5 :
:: deftheorem Def6 defines is_summable_set_by BHSP_6:def 6 :
definition
let c
1 be
RealUnitarySpace;
let c
2 be
Subset of c
1;
let c
3 be
Functional of c
1;
assume E8:
c
2 is_summable_set_by c
3
;
func sum_byfunc c
2,c
3 -> Real means :: BHSP_6:def 7
for b
1 being
Real holds
not ( b
1 > 0 & ( for b
2 being
finite Subset of a
1 holds
not ( not b
2 is
empty & b
2 c= a
2 & ( for b
3 being
finite Subset of a
1 holds
not ( b
2 c= b
3 & b
3 c= a
2 & not
abs (a4 - (setopfunc b3,the carrier of a1,REAL ,a3,addreal )) < b
1 ) ) ) ) );
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 )
end;
:: deftheorem Def7 defines sum_byfunc BHSP_6:def 7 :
theorem Th3: :: BHSP_6:3
theorem Th4: :: BHSP_6:4
theorem Th5: :: BHSP_6:5
theorem Th6: :: BHSP_6:6
theorem Th7: :: BHSP_6:7
theorem Th8: :: BHSP_6:8
theorem Th9: :: BHSP_6:9
theorem Th10: :: BHSP_6:10