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 Th54:
  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),
       m be Element of NAT,
       g,h be continuous PartFunc of REAL,the carrier of X
    st g = iter(Fredholm(G,a,b,y0),(m+1)).u
     & h = iter(Fredholm(G,a,b,y0),(m+1)).v holds
        for t be Real st t in ['a,b']
          holds ||. g/.t - h/.t .|| <= ((r*(t-a))|^(m+1))/((m+1)!) * ||.u-v.||
proof
   assume A1: a<=b & 0 < r
    & for y1,y2 be VECTOR of X holds ||.G/.y1-G/.y2.||<=r*||.y1-y2.||;
   set F = Fredholm(G,a,b,y0);
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 u1,v1 be VECTOR of R_NormSpace_of_ContinuousFunctions(['a,b'],X);
   defpred P[Nat] means
    for g,h be continuous PartFunc of REAL,the carrier of X
     st g = iter(F,($1+1)).u1 & h = iter(F,($1+1)).v1
    holds for t be Real st t in ['a,b']
           holds ||. g/.t - h/.t .||
              <= ((r*(t-a))|^($1+1))/(($1+1)!) * ||.u1-v1.||;
   reconsider Z0 = 0 as Element of NAT;
A4:P[0]
   proof
    let g,h be continuous PartFunc of REAL,the carrier of X;
    assume g = iter(F,( (0 qua Element of NAT ) + 1)).u1
         & h = iter(F,( (0 qua Element of NAT ) + 1)).v1; then
A6: g= F.u1 & h= F.v1 by FUNCT_7:70;
    for  t be Real st t in ['a,b'] holds
    ||. g/.t - h/.t .|| <= ((r*(t-a))|^(Z0 + 1) )
    /((Z0 + 1)!) * ||.u1-v1.|| by NEWTON:13,Th53,A1,A6;
    hence thesis;
   end;
A8:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume A9: P[k];
    let g,h be continuous PartFunc of REAL,the carrier of X;
    assume A10: g = iter(F,((k+1)+1)).u1 & h = iter(F,((k+1)+1)).v1;
    reconsider u=iter(F,(k+1)).u1, v=iter(F,(k+1)).v1 as
      VECTOR of R_NormSpace_of_ContinuousFunctions(['a,b'],X);
A11:dom iter(F,k+1) = the carrier of
      R_NormSpace_of_ContinuousFunctions(['a,b'],X) by FUNCT_2:def 1;
A12:iter(F,((k+1)+1)).u1 = (F*iter(F,k+1)).u1 by FUNCT_7:71
                        .= F.u by A11,FUNCT_1:13;
A13:iter(F,((k+1)+1)).v1 = (F*iter(F,k+1)).v1 by FUNCT_7:71
                        .= F.v by A11,FUNCT_1:13;
    consider f1,g1,Gf1 be continuous PartFunc of REAL,the carrier of X
      such that
A14: 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
A15: 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
A18:dom Gf1 =['a,b'] & dom Gf2 =['a,b'] by A14,A15,RELAT_1:27;
    reconsider Gf12 as continuous PartFunc of REAL,the carrier of X;
A20:['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
    let t be Real;
    assume A21: t in ['a,b']; then
A22:ex g be Real st t=g & a<=g & g <= b by A20;
a22:ex g be Element of REAL st t=g & a<=g & g <= b
    proof
     consider g be Real such that H1: t=g & a<=g & g <= b by A21,A20;
     reconsider g as Element of REAL by XREAL_0:def 1;
     take g;
     thus thesis by H1;
    end;
A23:dom Gf12 = dom Gf1 /\ dom Gf2 by VFUNCT_1:def 2
            .= ['a,b'] by A18; then
A24:dom ||.Gf12.|| = ['a,b'] by NORMSP_0:def 2;
A27:a in ['a,b'] by A20,A1;
    min(a,t) = a & max(a,t) = t by A22,XXREAL_0:def 9,def 10; then
A30:||.Gf12.|| is_integrable_on ['a,t'] &
      (||.Gf12.||) | ['a,t'] is bounded &
       ||.integral(Gf12,a,t).|| <= integral(||.Gf12.||,a,t)
                      by A1,A23,A27,A21,INTEGR21:22;
    ['a,t'] = [.a,t.] by A22,INTEGRA5:def 3; then
    consider rg be PartFunc of REAL,REAL such that
A31: dom rg = ['a,t']
   & (for x be Real st x in ['a,t']
          holds rg.x = r*(( (r*(x-a))|^(k+1) )/((k+1)!) * ||.u1-v1.|| ))
   & rg is continuous
   & rg is_integrable_on ['a,t'] & rg| ['a,t'] is bounded
   & integral(rg,a,t) = ((r*(t-a))|^((k+1)+1))
                          / (((k+1)+1)!) * ||.u1-v1.|| by Lm7,a22;
A32:['a,t'] c= ['a,b'] by A22,INTEGR19:1;
    for x be Real st x in ['a,t'] holds ||.Gf12.||.x <= rg.x
    proof
     let x be Real;
     assume A33: x in ['a,t']; then
A34: Gf12/.x =Gf1/.x -Gf2/.x by A23,A32,VFUNCT_1:def 2;
A35: Gf1/.x = (Gf1).x by A18,A32,A33,PARTFUN1:def 6
           .= G.(f1.x) by A32,A33,A18,A14,FUNCT_1:12
           .= G/.(f1/.x) by A32,A33,A14,PARTFUN1:def 6;
     Gf2/.x = (Gf2).x by A18,A32,A33,PARTFUN1:def 6
           .= G.(f2.x) by A32,A33,A18,A15,FUNCT_1:12
           .= G/.(f2/.x) by A32,A33,A15,PARTFUN1:def 6; then
A37: ||. Gf1/.x -Gf2/.x .|| <= r*||.(f1/.x)-(f2/.x).|| by A35,A1;
     r*||.(f1/.x)-(f2/.x).||
        <=r*(( (r*(x-a))|^(k+1) )/((k+1)!) * ||.u1-v1.||)
        by A1,XREAL_1:64,A9,A14,A15,A32,A33; then
     r*||.(f1/.x)-(f2/.x).|| <=rg.x by A33,A31; then
     ||. Gf1/.x -Gf2/.x .|| <=rg.x by A37,XXREAL_0:2;
     hence thesis by A24,A32,A33,NORMSP_0:def 2,A34;
    end; then
A38:integral(||.Gf12.||,a,t) <= integral(rg,a,t)
         by A30,A22,A24,A32,A31,ORDEQ_01:48;
    g/.t = y0+ integral(Gf1,a,t) & h/.t = y0+ integral(Gf2,a,t)
        by A14,A15,A21,A12,A10,A13; 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 A18,A27,A21,A1,INTEGR21:30;
    hence thesis by A38,A30,XXREAL_0:2,A31;
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(A4,A8);
   hence thesis;
end;
