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 Th49:
  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),
      g,h be continuous PartFunc of REAL,REAL-NS n
        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 REAL-NS n holds ||.G/.y1-G/.y2.||<=r*||.y1-y2.||;
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 u,v be VECTOR of R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n),
      g,h be continuous PartFunc of REAL,REAL-NS n;
  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,REAL-NS n 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 Def7,A1,A3;
  consider f2,g2,Gf2 be continuous PartFunc of REAL,REAL-NS n 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 Def7,A1,A3;
  set Gf12= Gf1 - Gf2;
A7: dom G = the carrier of REAL-NS n by FUNCT_2:def 1; then
  rng f1 c= dom G; then
A8: dom Gf1 =[' a,b '] by A5,RELAT_1:27;
  rng f2 c= dom G by A7; then
A9: dom Gf2 =[' a,b '] by A6,RELAT_1:27;
  reconsider Gf12 as continuous PartFunc of REAL,REAL-NS n;
A10: ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
  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;
A13: dom Gf12 = dom Gf1 /\ dom Gf2 by VFUNCT_1:def 2
             .= ['a,b'] by A8,A9;
A14: Gf12 is_integrable_on ['a,b'] by A13,Th33;
A15: Gf12| ['a,b'] is bounded  by A13,Th32;
     Gf12| (['a,b']) is continuous; then
A16: (||. Gf12 .||) | [' a,b '] is continuous by A13,NFCONT_3:22;
  [' a,b '] = dom (||. Gf12 .||) by A13,NORMSP_0:def 2; then
A17: (||. Gf12 .||) is_integrable_on [' a,b '] by A16,INTEGRA5:11;
A18: a in ['a,b'] by A10,A1;
  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'];
  A20: ['a,t'] c= [' a,b '] by A12,INTEGR19:1;
  A21: Gf12/.x =Gf1/.x -Gf2/.x by A13,A20,A19,VFUNCT_1:def 2;
  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;
  A23: Gf2/.x = (Gf2).x by A9,A20,A19,PARTFUN1:def 6
             .= G.(f2.x) by A20,A19,A9,A6,FUNCT_1:12
             .= G/.(f2/.x) by A20,A19,A6,PARTFUN1:def 6;
  A24: ||. Gf1/.x -Gf2/.x .||<=r*||.(f1/.x)-(f2/.x).|| by A22,A23,A1;
    ||.(f1/.x)-(f2/.x).|| <=||.u-v.|| by A20,A19,A5,A6,Th26; then
    r*||.(f1/.x)-(f2/.x).|| <=r*||.u-v.|| by A1,XREAL_1:64;
    hence thesis by A21,A24,XXREAL_0:2;
  end; then
A25: ||. integral(Gf12,a,t) .||<= ((r*||.u-v.|| ))* (t-a)
     by Th45,A1,A17,A14,A15,A13,A18,A11,A12;
A26: Gf1 is_integrable_on ['a,b'] by A8,Th33;
A27: Gf1| ['a,b'] is bounded by A8,Th32;
A28: Gf2 is_integrable_on ['a,b'] by A9,Th33;
A29: Gf2| ['a,b'] is bounded by A9,Th32;
A30: integral(Gf12,a,t) = integral(Gf1,a,t) - integral(Gf2,a,t)
       by A8,A9,A26,A27,A28,A29,A18,A11,A1,INTEGR19:50;
A31: g/.t = g1.t by A4,A11,A5,PARTFUN1:def 6
         .= y0 + integral(Gf1,a,t) by A5,A11;
A32: h/.t = g2.t by A4,A11,A6,PARTFUN1:def 6
         .= y0 + integral(Gf2,a,t) by A6,A11;
  g/.t - h/.t = (y0+ integral(Gf1,a,t) - y0 ) -integral(Gf2,a,t)
    by A31,A32,RLVECT_1:27
  .= ( 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 A30;
  hence thesis by A25;
end;
