reserve n,m for Nat,
  x,X,X1 for set,
  s,g,r,p for Real,
  S,T for RealNormSpace,
  f,f1,f2 for PartFunc of S, T,
  s1,s2,q1 for sequence of S,
  x0,x1, x2 for Point of S,
  Y for Subset of S;

theorem Th14:
  for X be RealBanachSpace
  for f be Function of X, X st f is Contraction of X
   ex xp be Point of X st f.xp=xp & for x be Point of X st f.x=x
  holds xp=x
proof
  let X be RealBanachSpace;
  set x0 = the Element of X;
  let f be Function of X,X;
  assume f is Contraction of X;
  then consider K be Real such that
A1: 0 < K and
A2: K < 1 and
A3: for x,y be Point of X holds ||.f.x - f.y.|| <=K*||.x - y.|| by Def3;
  deffunc G(set,set)=f.$2;
  consider g be Function such that
A4: dom g = NAT & g.0 = x0 & for n being Nat holds g.(n+1) = G(n,g.n)
  from NAT_1:sch 11;
  defpred P[Nat] means g.$1 in the carrier of X;
A5: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A6: P[k];
    g.(k+1)=f.(g.k) by A4;
    hence thesis by A6,FUNCT_2:5;
  end;
A7: P[0] by A4;
  for n be Nat holds P[n] from NAT_1:sch 2(A7,A5);
  then for n be object st n in NAT holds g.n in the carrier of X;
  then reconsider g as sequence of X by A4,FUNCT_2:3;
A8: now
    let n be Nat;
    ||. (g^\1).n - f. (lim g) .|| = ||. g.(n+1) - f. (lim g) .|| by NAT_1:def 3
      .= ||. f.(g.n)- f. (lim g) .|| by A4;
    hence ||. (g^\1).n - f. (lim g) .|| <= K* ||. g.n - lim g .|| by A3;
  end;
A9: for n be Nat holds ||.g.(n+1) - g.n .|| <= ||. g.1-g.0 .||*
  (K to_power n )
  proof
    defpred P[Nat] means
    ||.g.($1+1) - g.($1) .|| <= ||. g.1-g.0 .||* (K to_power $1 );
A10: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume P[k];
      then
A11:  K* ||. g.(k+1) - g.k .|| <= K*( ||. g.1-g.0 .||* (K to_power k ) )
      by A1,XREAL_1:64;
      ||.f.(g.(k+1)) - f.(g.k) .|| <= K* ||. g.(k+1) - g.k .|| by A3;
      then
      ||.f.(g.(k+1)) - f.(g.k) .|| <=||. g.1-g.0 .||*(K* (K to_power k ))
      by A11,XXREAL_0:2;
      then
A12:  ||.f.(g.(k+1)) - f.(g.k) .|| <=||. g.1-g.0 .||*((K to_power 1)* (K
      to_power k )) by POWER:25;
      g.(k+1) =f.(g.k) by A4;
      then
      ||.g.((k+1)+1) - g.(k+1) .|| = ||.f.(g.(k+1)) - f.(g.k) .|| by A4;
      hence thesis by A1,A12,POWER:27;
    end;
    ||.g.(0+1) - g.(0) .|| = ||. g.1-g.0 .|| * 1
      .= ||. g.1-g.0 .|| * (K to_power 0 ) by POWER:24;
    then
A13: P[0];
    for n be Nat holds P[n] from NAT_1:sch 2(A13,A10);
    hence thesis;
  end;
A14: for k, n be Nat holds ||.g.(n+k) - g.n .|| <= ||. g.1-g.0
  .||* ((K to_power n - K to_power (n+k)) /(1-K))
  proof
    defpred P[Nat] means
  for n be Nat holds ||.g.(n+$1)
    - g.n .|| <= ||. g.1-g.0 .||*( (K to_power n-K to_power (n+$1)) /(1-K));
A15: now
      let k be Nat such that
A16:  P[k];
      now
        let n be Nat;
        1- K <> 0 by A2;
        then
A17:    ||. g.1-g.0 .||*((K to_power n - K to_power (n+k)) /(1-K)) + ||.
g.1-g.0 .||*(K to_power (n+k)) = ||. g.1-g.0 .||*((K to_power n - K to_power (n
+k))/(1-K)) + ||. g.1-g.0 .||*(K to_power (n+k))*(1-K)/(1-K) by XCMPLX_1:89
          .= ||. g.1-g.0 .||*((K to_power n - K to_power (n+k))/(1-K)) + ||.
        g.1-g.0 .||*((K to_power (n+k))*(1-K))/(1-K)
          .= ||. g.1-g.0 .||*((K to_power n - K to_power (n+k))/(1-K)) + ||.
        g.1-g.0 .||*((K to_power (n+k)*(1-K))/(1-K)) by XCMPLX_1:74
          .= ||. g.1-g.0 .||*(((K to_power n - K to_power (n+k))/(1-K)) + ((
        K to_power (n+k)*(1-K))/(1-K)))
          .= ||. g.1-g.0 .||*( ((K to_power n - K to_power (n+k)) +(K
        to_power (n+k)*(1-K)))/(1-K) ) by XCMPLX_1:62
          .= ||. g.1-g.0 .||*((K to_power n- K*K to_power (n+k)) /(1-K))
          .= ||. g.1-g.0 .||*((K to_power n- (K to_power 1) *K to_power (n+k
        )) /(1-K)) by POWER:25
          .= ||. g.1-g.0 .||*((K to_power n - K to_power (n+k+1))/(1-K)) by A1,
POWER:27;
        ||.g.(n+k) - g.n .|| <= ||. g.1-g.0 .||* ((K to_power n - K
to_power (n+k)) / (1-K)) & ||.g.((n+k)+1) - g.(n+k) .|| <= ||. g.1-g.0 .||* (K
        to_power (n+k)) by A9,A16;
        then
        ||.g.(n+(k+1)) - g.n .|| <= ||.g.(n+(k+1)) - g.(n+k) .|| + ||.g.(
n+k) - g.n .|| & ||.g.(n+(k+1)) - g.(n+k) .|| + ||.g.(n+k) - g.n .|| <= ||. g.1
-g.0 .||* (K to_power (n+k)) + ||. g.1-g.0 .|| * ((K to_power n - K to_power (n
        +k)) /(1-K )) by NORMSP_1:10,XREAL_1:7;
        hence ||.g.(n+(k+1)) - g.n .|| <= ||. g.1-g.0 .||*((K to_power n - K
        to_power (n+(k+1)))/(1-K)) by A17,XXREAL_0:2;
      end;
      hence P[k+1];
    end;
    now
      let n be Nat;
      ||.g.(n+0) - g.n .||= ||.0.X.|| by RLVECT_1:15
        .=0 by NORMSP_1:1;
      hence ||.g.(n+0) - g.n .|| <= ||. g.1-g.0 .||* ((K to_power n-K to_power
      (n+0))/(1-K));
    end;
    then
A18: P[0];
    for k be Nat holds P[k] from NAT_1: sch 2(A18,A15);
    hence thesis;
  end;
A19: for k,n be Nat holds ||.g.(n+k) - g.n .|| <= ||. g.1-g.0 .||
  * (K to_power n /(1-K))
  proof
    let k be Nat;
    now
      let n be Nat;
      K to_power (n+k) > 0 by A1,POWER:34;
      then
A20:  K to_power n - K to_power (n+k) <= K to_power n - 0 by XREAL_1:13;
      1-K > 1-1 by A2,XREAL_1:15;
      then
      (K to_power n - K to_power (n+k)) /(1-K) <= (( K to_power n ) /(1-K
      )) by A20,XREAL_1:72;
      then
A21:  ||. g.1-g.0 .||*( (K to_power n - K to_power (n+k)) /(1-K) ) <= ||.
      g.1-g.0 .||*( ( K to_power n ) /(1-K) ) by XREAL_1:64;
      ||.g.(n+k) - g.n .|| <= ||. g.1-g.0 .||*( (K to_power n - K
      to_power (n+k)) /(1-K)) by A14;
      hence
      ||.g.(n+k) - g.n .|| <= ||. g.1-g.0 .||* ((K to_power n) /(1-K)) by A21,
XXREAL_0:2;
    end;
    hence thesis;
  end;
  now
    let e be Real such that
A22: e >0;
    e/2 > 0 by A22,XREAL_1:215;
    then consider n be Nat such that
A23: |. ||. g.1-g.0 .||/(1-K)* (K to_power n) .| < e/2 by A1,A2,Lm2;
    reconsider nn=n+1 as Nat;
    take nn;
    ||. g.1-g.0 .||/(1-K)* (K to_power n) <= |. ||. g.1-g.0 .||/(1-K)*
    (K to_power n) .| by ABSVALUE:4;
    then ||. g.1-g.0 .||/(1-K)* (K to_power n) < e/2 by A23,XXREAL_0:2;
    then
A24: ||. g.1-g.0 .|| *( (K to_power n)/(1-K)) < e/2 by XCMPLX_1:75;
    now
      let m,l be Nat such that
A25:  nn <= m and
A26:  nn <= l;
      n < m by A25,NAT_1:13;
      then consider k1 being Nat such that
A27:  n+k1 =m by NAT_1:10;
      n < l by A26,NAT_1:13;
      then consider k2 being Nat such that
A28:  n+k2 =l by NAT_1:10;
      reconsider k2 as Nat;
      ||.g.(n+k2) - g.n .|| <= ||. g.1-g.0 .||*( (K to_power n) /(1-K) )
      by A19;
      then
A29:  ||.g.l - g.n .|| < e/2 by A24,A28,XXREAL_0:2;
      reconsider k1 as Nat;
      ||.g.(n+k1) - g.n .|| <= ||. g.1-g.0 .||*( (K to_power n) /(1-K)) by A19;
      then ||.g.m - g.n .|| < e/2 by A24,A27,XXREAL_0:2;
      hence ||.g.l - g.m .|| < e by A22,A29,Lm1;
    end;
    hence
    for n, m be Nat st n >= nn & m >= nn holds ||.g.n - g.m.|| < e;
  end;
  then g is Cauchy_sequence_by_Norm by RSSPACE3:8;
  then
A30: g is convergent by LOPBAN_1:def 15;
  then
A31: K(#)||. g - lim g .|| is convergent by NORMSP_1:24,SEQ_2:7;
A32: lim (K(#)(||.g - lim g.||)) =K*lim ||.(g - lim g).|| by A30,NORMSP_1:24
,SEQ_2:8
    .=K*0 by A30,NORMSP_1:24
    .=0;
A33: for e be Real st e >0
   ex n be Nat st for m be Nat
  st n<=m holds ||. (g^\1).m - f. (lim g) .|| <e
  proof
    let e be Real;
    assume e >0;
    then consider n be Nat such that
A34: for m be Nat st n<=m holds |. (K(#)||. g - lim g .||
    ).m-0 .| <e by A31,A32,SEQ_2:def 7;
    take n;
    now
      let m be Nat;
      assume n<=m;
      then |. (K(#)||. g - lim g .||).m-0 .|<e by A34;
      then |. K*||. g - lim g .||.m.| < e by SEQ_1:9;
      then |. K*||. (g - lim g).m .||.| < e by NORMSP_0:def 4;
      then
A35:  |. K*||. g.m - lim g .||.| < e by NORMSP_1:def 4;
      K*||. g.m - lim g .|| <=|. K*||. g.m - lim g .||.| by ABSVALUE:4;
      then
A36:  K*||. g.m - lim g .|| < e by A35,XXREAL_0:2;
      ||. (g^\1).m - f. (lim g) .|| <= K* ||. g.m - lim g .|| by A8;
      hence ||. (g^\1).m - f. (lim g) .|| < e by A36,XXREAL_0:2;
    end;
    hence thesis;
  end;
  take xp=lim g;
A37: g^\1 is convergent & lim (g^\1) = lim g by A30,LOPBAN_3:9;
  then
A38: lim g = f. (lim g) by A33,NORMSP_1:def 7;
  now
    let x be Point of X such that
A39: f.x=x;
A40: for k be Nat holds ||.x-xp.|| <= ||.x-xp.||*(K to_power k)
    proof
      defpred P[Nat] means
   ||.x-xp .|| <= ||.x-xp.||*(K to_power $1);
A41:  for k be Nat st P[k] holds P[k+1]
      proof
        let k be Nat;
        assume P[k];
        then
A42:    K* ||. x-xp .|| <= K*( ||.x-xp.||*(K to_power k) ) by A1,XREAL_1:64;
        ||.f.x - f.xp .|| <= K* ||. x-xp .|| by A3;
        then ||.f.x - f.xp .|| <= ||. x-xp .||*(K* (K to_power k )) by A42,
XXREAL_0:2;
        then ||.f.x - f.xp .|| <= ||. x-xp .||*((K to_power 1)* (K to_power k
        )) by POWER:25;
        hence thesis by A1,A38,A39,POWER:27;
      end;
      ||.x-xp .|| = ||. x-xp .|| * 1
        .= ||. x-xp .|| * (K to_power 0 ) by POWER:24;
      then
A43:  P[0];
      for n be Nat holds P[n] from NAT_1:sch 2(A43,A41);
      hence thesis;
    end;
    for e be Real st 0 <e holds ||.x-xp.|| < e
    proof
      let e be Real;
      assume 0 < e;
      then consider n be Nat such that
A44:  |.||.x-xp.||*(K to_power n) .| < e by A1,A2,Lm2;
      ||.x-xp.||*(K to_power n) <=|.||.x-xp.||*(K to_power n).| by ABSVALUE:4;
      then
A45:  ||.x-xp.||*(K to_power n) <e by A44,XXREAL_0:2;
      ||.x-xp.|| <= ||.x-xp.||*(K to_power n) by A40;
      hence thesis by A45,XXREAL_0:2;
    end;
    hence x=xp by Lm1;
  end;
  hence thesis by A37,A33,NORMSP_1:def 7;
end;
