let n be Element of NAT ; :: thesis: for a, b being Real
for f being PartFunc of REAL,(REAL-NS n) ex F being PartFunc of REAL,(REAL-NS n) st
( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds
F . x = integral (f,a,x) ) )
let a, b be Real; :: thesis: for f being PartFunc of REAL,(REAL-NS n) ex F being PartFunc of REAL,(REAL-NS n) st
( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds
F . x = integral (f,a,x) ) )
deffunc H₁( Real) -> Element of the carrier of (REAL-NS n) = 0. (REAL-NS n);
let f be PartFunc of REAL,(REAL-NS n); :: thesis: ex F being PartFunc of REAL,(REAL-NS n) st
( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds
F . x = integral (f,a,x) ) )
defpred S₁[ set ] means $1 in ].a,b.[;
deffunc H₂( Real) -> Element of the carrier of (REAL-NS n) = integral (f,a,$1);
consider F being Function such that
A1: ( dom F = REAL & ( for x being Element of REAL holds
( ( S₁[x] implies F . x = H₂(x) ) & ( not S₁[x] implies F . x = H₁(x) ) ) ) ) from PARTFUN1:sch 4();

now :: thesis: for y being object st y in rng F holds
y in the carrier of (REAL-NS n)

let y be object ; :: thesis: ( y in rng F implies y in the carrier of (REAL-NS n) )
assume y in rng F ; :: thesis: y in the carrier of (REAL-NS n)
then consider x being object such that
A2: x in dom F and
A3: y = F . x by FUNCT_1:def 3;
reconsider x = x as Element of REAL by A1, A2;

A4: now :: thesis: ( not x in ].a,b.[ implies y in the carrier of (REAL-NS n) )

assume not x in ].a,b.[ ; :: thesis: y in the carrier of (REAL-NS n)
then F . x = 0. (REAL-NS n) by A1;
hence y in the carrier of (REAL-NS n) by A3; :: thesis: verum

end;

now :: thesis: ( x in ].a,b.[ implies y in the carrier of (REAL-NS n) )

assume x in ].a,b.[ ; :: thesis: y in the carrier of (REAL-NS n)
then F . x = integral (f,a,x) by A1;
hence y in the carrier of (REAL-NS n) by A3; :: thesis: verum

end;

hence y in the carrier of (REAL-NS n) by A4; :: thesis: verum

end;

then rng F c= the carrier of (REAL-NS n) ;
then reconsider F = F as PartFunc of REAL,(REAL-NS n) by A1, RELSET_1:4;
take F ; :: thesis: ( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds
F . x = integral (f,a,x) ) )
thus ( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds
F . x = integral (f,a,x) ) ) by A1; :: thesis: verum