 reserve a,Z1,Z2,Z3 for set,
         x,y,z for object,
         k for Nat;
 reserve S for RelStr;
 reserve P,Q for non empty flat Poset;
 reserve p,p1,p2 for Element of P;
 reserve K for non empty Chain of P;
 reserve X,Y for non empty set;
 reserve D for Subset of X;
 reserve I for Function of X,Y;
 reserve J for Function of [:X,Y:], Y;
 reserve E for Function of X,X;

theorem Threcursive02:
  ex f being set st f in ConFuncs(FlatPoset(X),FlatPoset(Y)) &
     f = RecFunc01(f,E,I,J,D)
  proof
    set FX = FlatPoset(X);
    set FY = FlatPoset(Y);
    set FlatC = FlatConF(X,Y);
    set CFXY = ConFuncs(FX,FY);
    set CRFXY = ConRelat(FX,FY);
    consider W be continuous Function of FlatC,FlatC such that
A4: for f being Element of CFXY
       holds W.f = RecFunc01(f,E,I,J,D) by Threcursive01;
    reconsider W as monotone Function of FlatC,FlatC;
    reconsider f = least_fix_point(W) as Element of FlatC;
A5: f is_a_fixpoint_of W by POSET_1:def 5;
A6: f = W.f by A5,ABIAN:def 3;
    take f;
    thus thesis by A4,A6;
  end;
