
theorem
  for R being non empty well_founded RelStr, V being non empty set, H
being Function of [:the carrier of R, PFuncs(the carrier of R, V):], V, F1, F2
being Function of the carrier of R, V st F1 is_recursively_expressed_by H & F2
  is_recursively_expressed_by H holds F1 = F2
proof
  let R be non empty well_founded RelStr, V be non empty set;
  set c = the carrier of R, r = the InternalRel of R;
  let H be Function of [:c, PFuncs(c, V):], V, F1, F2 be Function of c, V;
  assume that
A1: F1 is_recursively_expressed_by H and
A2: F2 is_recursively_expressed_by H;
  defpred P[set] means F1.$1 <> F2.$1;
A3: dom F2 = c by FUNCT_2:def 1;
  assume F1 <> F2;
  then consider x being Element of c such that
A4: F1.x <> F2.x by FUNCT_2:63;
  reconsider x as Element of R;
A5: R is well_founded;
A6: P[x] by A4;
  consider x0 being Element of R such that
A7: P[x0] and
A8: not ex y being Element of R st x0 <> y & P[y] & [y,x0] in r from
  WFMin(A6, A5);
A9: dom F1 = c by FUNCT_2:def 1;
  F1 | r-Seg x0 = F2 | r-Seg x0
  proof
    set fr = F1 | r-Seg x0, gr = F2 | r-Seg x0;
    assume
A10: not thesis;
A11: dom fr = r-Seg x0 by A9,Th3,RELAT_1:62;
    dom gr = r-Seg x0 by A3,Th3,RELAT_1:62;
    then consider x1 being object such that
A12: x1 in dom fr and
A13: fr.x1 <> gr.x1 by A11,A10;
A14: [x1, x0] in r by A11,A12,WELLORD1:1;
    reconsider x1 as Element of R by A12;
A15: x1 <> x0 by A11,A12,WELLORD1:1;
    fr.x1 = F1.x1 & gr.x1 = F2.x1 by A11,A12,FUNCT_1:49;
    hence contradiction by A8,A13,A14,A15;
  end;
  then F1.x0 = H.(x0, F2 | r-Seg x0) by A1
    .= F2.x0 by A2;
  hence contradiction by A7;
end;
