 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
  E is_well_founded_with_minimal_set D implies
    ex f being Function of X,Y st
       for x being Element of X holds
         (x in D implies f.x = I.x) &
         (not x in D implies f.x = J.[x,f.(E.x)])
  proof
    assume E is_well_founded_with_minimal_set D;
    then consider f being Function of X,Y such that
A1: for x being Element of X holds f.x = BaseFunc01(x,f.(E.x),I,J,D)
                                    by Lemrecursive04;
    take f;
    let x be Element of X;
    f.x = BaseFunc01(x,f.(E.x),I,J,D) by A1;
    hence thesis by DefBaseFunc01;
  end;
