reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;
reserve
  Z for open Subset of REAL,
 y0 for VECTOR of REAL-NS n,
  G for Function of REAL-NS n,REAL-NS n;

theorem Th50:
  a <= b & 0 < r &
  (for y1,y2 be VECTOR of REAL-NS n holds ||.G/.y1-G/.y2.|| <= r*||.y1-y2.||)
  implies
  for u,v be VECTOR of R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n),
        m be Element of NAT,
      g,h be continuous PartFunc of REAL,REAL-NS n
        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 REAL-NS n holds ||.G/.y1-G/.y2.||<=r*||.y1-y2.||;
  set F = Fredholm(G,a,b,y0);
A2: dom G = the carrier of REAL-NS n by FUNCT_2:def 1;
  for x1,x2 be Point of REAL-NS n
   st x1 in (the carrier of REAL-NS n)
    & x2 in (the carrier of REAL-NS n) holds
  ||.G/.x1-G/.x2.||<=r*||.x1-x2.|| by A1; then
  G is_Lipschitzian_on the carrier of (REAL-NS n) by A1,A2,NFCONT_1:def 9; 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 '],REAL-NS n);
  defpred P[Nat] means
  for g,h be continuous PartFunc of REAL,REAL-NS n
    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,REAL-NS n;
    assume A5: g = iter(F,( (0 qua Element of NAT ) + 1)).u1
    & h = iter(F,( (0 qua Element of NAT ) + 1)).v1;
  A6: g= F.u1 by A5,FUNCT_7:70;
  A7: h= F.v1 by A5,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,Th49,A1,A6,A7;
    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,REAL-NS n;
    assume A10:
      g = iter(F,((k+1)+1)).u1 & h = iter(F,((k+1)+1)).v1;
    reconsider u=iter(F,(k+1)).u1 as
      VECTOR of R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n);
    reconsider v=iter(F,(k+1)).v1 as
      VECTOR of R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n);
  A11: dom(iter(F,k+1)) = the carrier of
    R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n) 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,REAL-NS n 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 Def7,A1,A3;
    consider f2,g2,Gf2 be continuous PartFunc of REAL,REAL-NS n 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 Def7,A1,A3;
    set Gf12= Gf1 - Gf2;
  A16: for t be Real st t in [' a,b ']
         holds ||. f1/.t - f2/.t .||
               <= ( (r*(t-a))|^(k+1) )/((k+1)!) * ||.u1-v1.|| by A9,A14,A15;
  A17: dom G = the carrier of REAL-NS n by FUNCT_2:def 1; then
    rng f1 c= dom G; then
  A18: dom Gf1 =[' a,b '] by A14,RELAT_1:27;
    rng f2 c= dom G by A17; then
  A19: dom Gf2 =[' a,b ']  by A15,RELAT_1:27;
    reconsider Gf12 as continuous PartFunc of REAL,REAL-NS n;
  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;
  A23: ex g be Element of REAL st t=g & a<=g & g <= b
       proof
         consider g be Real such that A24: 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 A24;
       end;
  A25: dom Gf12 = dom Gf1 /\ dom Gf2 by VFUNCT_1:def 2
               .= ['a,b'] by A18,A19; then
  A26: dom (||. Gf12 .||) = ['a,b'] by NORMSP_0:def 2;
  A27: Gf12 is_integrable_on ['a,b'] by A25,Th33;
  A28: Gf12| ['a,b'] is bounded by A25,Th32;
  A29: a in ['a,b'] by A20,A1;
    Gf12 | ['a,b'] is continuous; then
  A30: (||. Gf12 .||) | [' a,b '] is continuous by A25,NFCONT_3:22;
    [' a,b '] = dom (||. Gf12 .||) by A25,NORMSP_0:def 2; then
  A31: ||. Gf12 .|| is_integrable_on [' a,b '] by A30,INTEGRA5:11;
    min(a,t) = a & max(a,t) = t by A22,XXREAL_0:def 9,def 10; then
  A32: ||. Gf12 .|| is_integrable_on ['a,t'] &
      (||. Gf12 .||) | ['a,t'] is bounded &
       ||. integral(Gf12,a,t) .||
         <= integral((||. Gf12 .||),a,t) by A1,A27,A28,A25,A29,A21,A31,Th44;
A33: k+1 is non zero Element of NAT by ORDINAL1:def 12;
    consider rg be PartFunc of REAL,REAL such that
  A34: 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 Lm12,A23,A33;
  A35: ['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 A36: x in ['a,t'];
    A37: Gf12/.x =Gf1/.x -Gf2/.x by A25,A35,A36,VFUNCT_1:def 2;
    A38: Gf1/.x = (Gf1).x by A18,A35,A36,PARTFUN1:def 6
               .= G.(f1.x) by A35,A36,A18,A14,FUNCT_1:12
               .= G/.(f1/.x) by A35,A36,A14,PARTFUN1:def 6;
    A39: Gf2/.x = (Gf2).x by A19,A35,A36,PARTFUN1:def 6
               .= G.(f2.x) by A35,A36,A19,A15,FUNCT_1:12
               .= G/.(f2/.x) by A35,A36,A15,PARTFUN1:def 6;
    A40: ||. Gf1/.x -Gf2/.x .|| <= r*||.(f1/.x)-(f2/.x).|| by A38,A39,A1;
      r*||.(f1/.x)-(f2/.x).||
        <=r*(( (r*(x-a))|^(k+1) )/((k+1)!) * ||.u1-v1.||)
        by A1,XREAL_1:64,A16,A35,A36; then
      r*||.(f1/.x)-(f2/.x).|| <=rg.x by A36,A34; then
      ||. Gf1/.x -Gf2/.x .|| <=rg.x by A40,XXREAL_0:2;
      hence thesis by A26,A35,A36,NORMSP_0:def 2,A37;
    end; then
  A41: integral((||. Gf12 .||),a,t) <= integral(rg,a,t)
    by A32,A22,A26,A35,A34,Th48;
  A42: Gf1 is_integrable_on ['a,b'] by A18,Th33;
  A43: Gf1| ['a,b'] is bounded by A18,Th32;
  A44: Gf2 is_integrable_on ['a,b'] by A19,Th33;
  A45: Gf2| ['a,b'] is bounded by A19,Th32;
  A46: integral(Gf12,a,t) = integral(Gf1,a,t) - integral(Gf2,a,t)
     by A18,A19,A42,A43,A44,A45,A29,A21,A1,INTEGR19:50;
  A47: g/.t = g1.t by A12,A10,A21,A14,PARTFUN1:def 6
           .= y0+ integral(Gf1,a,t) by A14,A21;
  A48: h/.t = g2.t by A13,A10,A21,A15,PARTFUN1:def 6
           .= y0+ integral(Gf2,a,t) by A15,A21;
    g/.t - h/.t = (y0+ integral(Gf1,a,t) - y0 ) -integral(Gf2,a,t)
                  by RLVECT_1:27,A47,A48
               .= ( integral(Gf1,a,t) + (y0 - y0) ) -integral(Gf2,a,t)
                  by RLVECT_1:28
               .= ( integral(Gf1,a,t) + 0.(REAL-NS n) ) -integral(Gf2,a,t)
                  by RLVECT_1:15
               .= integral(Gf12,a,t) by A46;
    hence thesis by A41,A32,XXREAL_0:2,A34;
  end;
  for k be Nat holds P[k] from NAT_1:sch 2(A4,A8);
  hence thesis;
end;
