 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;
 reserve D for Subset of X;
 reserve I,I1,I2 for Function of X,Y;
 reserve J,J1,J2 for Function of [:X,Y,Y:], Y;
 reserve E1,E2 for Function of X,X;

theorem Threcursive0201:
  ex f,g being set st
     f in ConFuncs(FlatPoset(X),FlatPoset(Y)) &
     g in ConFuncs(FlatPoset(X),FlatPoset(Y)) &
     f = RecFunc02(f,g,E1,E2,I1,J1,D) &
     g = RecFunc02(f,g,E1,E2,I2,J2,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);
    set FlatC2 = [:FlatC,FlatC:];
    set CFXY2 = [:CFXY,CFXY:];
    set CRFXY2 = ["CRFXY,CRFXY"];
A4: the carrier of FlatC2 = CFXY2 &
         the InternalRel of FlatC2 = CRFXY2 by YELLOW_3:def 2;
    consider W1 be continuous Function of FlatC2,FlatC such that
A5: for h being set st h in CFXY2
                 holds W1.h = RecFunc02(h`1,h`2,E1,E2,I1,J1,D)
                                        by Threcursive0101;
    consider W2 be continuous Function of FlatC2,FlatC such that
A6: for h being set st h in CFXY2
                 holds W2.h = RecFunc02(h`1,h`2,E1,E2,I2,J2,D)
                                        by Threcursive0101;
    reconsider W = <:W1,W2:> as continuous Function of FlatC2,FlatC2
      by YELLOW_3:def 2;
    reconsider h = least_fix_point(W) as Element of FlatC2;
    h is_a_fixpoint_of W by POSET_1:def 5; then
A7: h = W.h by ABIAN:def 3;
A8: W1.h = RecFunc02(h`1,h`2,E1,E2,I1,J1,D) &
      W2.h = RecFunc02(h`1,h`2,E1,E2,I2,J2,D) by A4,A5,A6;
    consider f,g being Element of FlatC such that
A9: h = [f,g] by ThProdPoset01;
    take f,g;
    dom W = (the carrier of FlatC2) by FUNCT_2:def 1; then
    [f,g] = [RecFunc02(f,g,E1,E2,I1,J1,D),
             RecFunc02(f,g,E1,E2,I2,J2,D) ] by FUNCT_3:def 7,A7,A8,A9;
    hence thesis by XTUPLE_0:1;
  end;
