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 Th53:
a <= b & 0 < r &
(for y1,y2 be VECTOR of X holds ||.G/.y1-G/.y2.|| <= r*||.y1-y2.||)
 implies
  for u,v be VECTOR of R_NormSpace_of_ContinuousFunctions(['a,b'],X),
      g,h be continuous PartFunc of REAL,the carrier of X
        st g= Fredholm(G,a,b,y0).u & h= Fredholm(G,a,b,y0).v holds
         for t be Real st t in ['a,b'] holds
          ||. g/.t - h/.t .|| <= r*(t-a)*||.u-v.||
proof
   assume A1: a<=b & 0 < r
    & for y1, y2 be VECTOR of X holds ||.G/.y1-G/.y2.||<=r*||.y1-y2.||;
A2:dom G = the carrier of X by FUNCT_2:def 1;
   for x1,x2 be Point of X
     st x1 in (the carrier of X) & x2 in (the carrier of X)
    holds ||.G/.x1-G/.x2.||<=r*||.x1-x2.|| by A1; then
   G is_Lipschitzian_on the carrier of X by A1,FUNCT_2:def 1; then
A3:G is_continuous_on dom G by A2,NFCONT_1:45;
   let u,v be VECTOR of R_NormSpace_of_ContinuousFunctions(['a,b'],X),
       g,h be continuous PartFunc of REAL,the carrier of X;
   assume A4: g= Fredholm(G,a,b,y0).u & h= Fredholm(G,a,b,y0).v;
   set F= Fredholm(G,a,b,y0);
   consider f1,g1,Gf1 be continuous PartFunc of REAL,the carrier of X
     such that
A5: u=f1 & F.u = g1 & dom f1 =['a,b'] & dom g1 =['a,b'] & Gf1 = G*f1
  & for t be Real st t in ['a,b']
        holds g1/.t = y0+ integral(Gf1,a,t) by Def8,A1,A3;
   consider f2,g2,Gf2 be continuous PartFunc of REAL,the carrier of X
     such that
A6: v=f2 & F.v = g2 & dom f2 =['a,b'] & dom g2 =['a,b'] & Gf2 = G*f2
  & for t be Real st t in ['a,b']
        holds g2/.t = y0+ integral(Gf2,a,t) by Def8,A1,A3;
   set Gf12= Gf1 - Gf2;
   dom G = the carrier of X by FUNCT_2:def 1; then
   rng f1 c= dom G & rng f2 c= dom G; then
A8:dom Gf1 =['a,b'] & dom Gf2 =['a,b'] by A5,A6,RELAT_1:27;
   reconsider Gf12 as continuous PartFunc of REAL,the carrier of X;
A10:['a,b'] = [.a,b.] by A1,INTEGRA5:def 3; then
A18:a in ['a,b'] by A1;
   let t be Real;
   assume A11: t in ['a,b']; then
A12:ex g be Real st t=g & a<=g & g <= b by A10; then
A20:['a,t'] c= ['a,b'] by INTEGR19:1;
X1:['a,t'] = [.a,t.] by A12,INTEGRA5:def 3;
A13:dom Gf12 = dom Gf1 /\ dom Gf2 by VFUNCT_1:def 2;
   for x be Real st x in ['a,t'] holds ||. Gf12/.x .|| <= r*||.u-v.||
   proof
    let x be Real;
    assume A19: x in ['a,t']; then
A21:Gf12/.x =Gf1/.x -Gf2/.x by A8,A13,A20,VFUNCT_1:def 2;
A24:r*||.(f1/.x)-(f2/.x).|| <=r*||.u-v.||
                     by A1,XREAL_1:64,A19,A20,A5,A6,ORDEQ_01:26;
A22:Gf1/.x = (Gf1).x by A8,A20,A19,PARTFUN1:def 6
          .= G.(f1.x) by A20,A19,A8,A5,FUNCT_1:12
          .= G/.(f1/.x) by A20,A19,A5,PARTFUN1:def 6;
    Gf2/.x = (Gf2).x by A8,A20,A19,PARTFUN1:def 6
          .= G.(f2.x) by A20,A19,A8,A6,FUNCT_1:12
          .= G/.(f2/.x) by A20,A19,A6,PARTFUN1:def 6; then
    ||. Gf1/.x -Gf2/.x .||<=r*||.(f1/.x)-(f2/.x).|| by A22,A1;
    hence thesis by A21,A24,XXREAL_0:2;
   end; then
A25: ||. integral(Gf12,a,t) .||<= r*||.u-v.||*(t-a)
          by Lm14a,X1,A8,A13,A18,A10,A11,A12;
   g/.t = y0 + integral(Gf1,a,t) & h/.t = y0 + integral(Gf2,a,t)
      by A4,A5,A6,A11; then
   g/.t - h/.t
     = (y0+ integral(Gf1,a,t) - y0 ) -integral(Gf2,a,t) by RLVECT_1:27
    .= integral(Gf1,a,t) + (y0 - y0) -integral(Gf2,a,t) by RLVECT_1:28
    .= integral(Gf1,a,t) + 0.X -integral(Gf2,a,t) by RLVECT_1:15
    .= integral(Gf12,a,t) by A8,A18,A11,A1,INTEGR21:30;
   hence thesis by A25;
end;
