reserve Y for RealNormSpace;
reserve X,Y for RealBanachSpace;
reserve Z for open Subset of REAL;
reserve a,b,c,d,e,r,x0 for Real;
reserve y0 for VECTOR of X;
reserve G for Function of X,X;

theorem Th40a:
for f be continuous PartFunc of REAL,the carrier of X,
    g be PartFunc of REAL,the carrier of X
  st a <= b & dom f = [.a,b.]
   & for t be Real st t in [.a,b.] holds g/.t = y0 + integral(f,a,t) holds
   g/.a = y0
proof
   let f be continuous PartFunc of REAL,the carrier of X,
       g be PartFunc of REAL,the carrier of X;
   assume that
A1:a <= b and
A2:dom f = [.a,b.] and
A6:for t be Real st t in [.a,b.] holds g/.t = y0+ integral(f,a,t);
A4:['a,b'] = [.a,b.] by A1,INTEGRA5:def 3; then
   a in ['a,b'] by A1; then
   integral(f,a,a) = 0.X & g/.a = y0 + integral(f,a,a) by A2,A6,A4,Lm00;
   hence g/.a = y0;
end;
