reserve X,X1 for set,
  r,s for Real,
  z for Complex,
  RNS for RealNormSpace,
  CNS, CNS1,CNS2 for ComplexNormSpace;

theorem
  for X be ComplexBanachSpace for f be Function of X,X st ex n0 be
Element of NAT st iter(f,n0) is Contraction of X holds 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 ComplexBanachSpace;
  let f be Function of X, X;
  given n0 be Element of NAT such that
A1: iter(f,n0) is Contraction of X;
  consider xp be Point of X such that
A2: (iter(f,n0)).xp=xp and
A3: for x be Point of X st iter(f,n0).x=x holds xp=x by A1,Th40;
A4: now
    let x be Point of X such that
A5: f.x=x;
    for n be Element of NAT holds (iter(f,n)).x=x
    proof
      defpred P[Nat] means (iter(f,$1)).x=x;
A6:   now
        let n be Nat such that
A7:     P[n];
A8:     iter(f,n) is Function of X,X by FUNCT_7:83;
        (iter(f,n+1)).x =(f*iter(f,n)).x by FUNCT_7:71
          .=x by A5,A7,A8,FUNCT_2:15;
        hence P[n+1];
      end;
      (iter(f,0)).x=(id the carrier of X).x by FUNCT_7:84
        .=x;
      then
A9:   P[0];
      for n be Nat holds P[n] from NAT_1:sch 2(A9,A6);
      hence thesis;
    end;
    then (iter(f,n0)).x=x;
    hence xp=x by A3;
  end;
A10: iter(f,n0) is Function of X,X by FUNCT_7:83;
  (iter(f,n0)).(f.xp) =(iter(f,n0)*f).xp by FUNCT_2:15
    .=(iter(f,n0+1)).xp by FUNCT_7:69
    .=(f*iter(f,n0)).xp by FUNCT_7:71
    .=f.xp by A2,A10,FUNCT_2:15;
  then f.xp=xp by A3;
  hence thesis by A4;
end;
