 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 Threcursive05:
  E1,E2 is_well_founded_with_minimal_set D implies
    ex f,g being Function of X,Y st
       for x being Element of X holds
         (x in D implies f.x = I1.x & g.x = I2.x) &
         (not x in D implies f.x = J1.[x,f.(E1.x),g.(E2.x)] &
                             g.x = J2.[x,f.(E1.x),g.(E2.x)])
  proof
    assume E1,E2 is_well_founded_with_minimal_set D;
    then consider f,g being Function of X,Y such that
A1: for x being Element of X holds
        f.x = BaseFunc02(x,f.(E1.x),g.(E2.x),I1,J1,D) &
        g.x = BaseFunc02(x,f.(E1.x),g.(E2.x),I2,J2,D) by Lemrecursive0401;
    take f,g;
    let x be Element of X;
    f.x = BaseFunc02(x,f.(E1.x),g.(E2.x),I1,J1,D) &
    g.x = BaseFunc02(x,f.(E1.x),g.(E2.x),I2,J2,D) by A1;
    hence thesis by DefBaseFunc02;
  end;
