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 Th47:
for y,Gf be continuous PartFunc of REAL,the carrier of X,
    g be PartFunc of REAL,the carrier of X
  st a <= b & Z = ].a,b.[
   & dom y = [.a,b.] & dom g = [.a,b.] & dom Gf = [.a,b.]
   & y is_differentiable_on Z
   & y/.a = y0
   & (for t be Real st t in Z holds diff(y,t) = Gf/.t)
   & (for t be Real st t in [.a,b.] holds g/.t = y0 + integral(Gf,a,t) )
holds y=g
proof
   let y,Gf be continuous PartFunc of REAL,the carrier of X,
       g be PartFunc of REAL,the carrier of X;
   assume A1: a <= b & Z = ].a,b.[
      & dom y = [.a,b.] & dom g =[.a,b.] & dom Gf =[.a,b.]
      & y is_differentiable_on Z
      & y/.a = y0
      & (for t be Real st t in Z holds diff(y,t) = Gf/.t)
      & (for t be Real st t in [.a,b.] holds g/.t = y0+ integral(Gf,a,t));
   then
A2:g is continuous & g/.a=y0 &
g is_differentiable_on Z
 & for t be Real st t in Z holds diff(g,t) = Gf/.t by Th40,Th40a;
   reconsider h=y-g as continuous PartFunc of REAL,the carrier of X by A2;
A5:dom h = [.a,b.] /\ [.a,b.] by A1,VFUNCT_1:def 2; then
A7:h is_differentiable_on ].a,b.[ by A1,A2,NDIFF_3:18;
A8:now let x be Real;
    assume A9: x in ].a,b.[; then
A10:diff(y,x) = Gf/.x & diff(g,x) = Gf/.x by A1,Th40;
    thus diff(h,x) = ((y-g)`|(].a,b.[)).x by A9,A7,NDIFF_3:def 6
      .= Gf/.x - Gf/.x by A10,A1,A2,A5,A9,NDIFF_3:18
      .= 0.X by RLVECT_1:15;
   end;
   for x be Real st x in [.a,b.] holds h is_continuous_in x
                              by A5,NFCONT_3:def 2; then
A12:
   h|(].a,b.[) is constant by A1,Th45,A5,A2,NDIFF_3:18,A8;
A13:for x being Real st x in dom h holds h/.x = 0.X
   proof
    let x be Real;
    assume A14: x in dom h;
A15:a in dom h by A5,A1;
    thus h/.x = h/.a by A14,Th46,A12,A5
      .= y0-y0 by A2,A1,A15,VFUNCT_1:def 2
      .= 0.X by RLVECT_1:15;
   end;
   for x being Element of REAL st x in dom y holds y.x = g.x
   proof
    let x be Element of REAL;
    assume A16: x in dom y; then
    0.X = h/.x by A13,A1,A5
       .= y/.x -g/.x by A16,A1,A5,VFUNCT_1:def 2; then
A17:y/.x = g/.x by RLVECT_1:21;
    thus y.x = y/.x by A16,PARTFUN1:def 6
            .= g.x by A17,A16,A1,PARTFUN1:def 6;
   end;
   hence thesis by A1,PARTFUN1:5;
end;
