reserve x,y,z,x1,x2,x3,x4,y1,y2,s for Variable,
  M for non empty set,
  a,b for set,
  i,j,k for Element of NAT,
  m,m1,m2,m3,m4 for Element of M,
  H,H1,H2 for ZF-formula,
  v,v9,v1,v2 for Function of VAR,M;

theorem Th14:
  not x.0 in Free H & M,v |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '='
x.0))) & not x in variables_in H & not y in Free H & x <> x.0 & x <> x.3 & x <>
x.4 implies not x.0 in Free (H/(y,x)) & M,v/(x,v.y) |= All(x.3,Ex(x.0,All(x.4,H
  /(y,x) <=> x.4 '=' x.0))) & def_func'(H,v) = def_func'(H/(y,x),v/(x,v.y))
proof
A1: x.0 <> x.4 by ZF_LANG1:76;
A2: x.3 <> x.4 by ZF_LANG1:76;
  set F = H/(y,x), f = v/(x,v.y);
  assume that
A3: not x.0 in Free H and
A4: M,v |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0))) and
A5: not x in variables_in H and
A6: not y in Free H and
A7: x <> x.0 and
A8: x <> x.3 and
A9: x <> x.4;
  Free F c= variables_in F & not x.0 in variables_in F or not x.0 in {x} &
  not x.0 in Free H \ {y} by A3,A7,TARSKI:def 1,XBOOLE_0:def 5;
  then
  not x.0 in Free F or Free F c= (Free H \ {y}) \/ {x} & not x.0 in (Free
  H \ {y}) \/ {x} by Th1,XBOOLE_0:def 3;
  hence
A10: not x.0 in Free F;
A11: x.0 <> x.3 by ZF_LANG1:76;
  now
    let m3;
    M,v/(x.3,m3) |= Ex(x.0,All(x.4,H <=> x.4 '=' x.0)) by A4,ZF_LANG1:71;
    then consider m such that
A12: M,v/(x.3,m3)/(x.0,m) |= All(x.4,H <=> x.4 '=' x.0) by ZF_LANG1:73;
    set f1 = f/(x.3,m3)/(x.0,m);
    now
      let m4;
A13:  v/(x.3,m3)/(x.0,m)/(x.4,m4).(x.4) = m4 by FUNCT_7:128;
A14:  v/(x.3,m3)/(x.0,m)/(x.4,m4).(x.0) = v/(x.3,m3)/(x.0,m).(x.0) by
FUNCT_7:32,ZF_LANG1:76;
A15:  f1/(x.4,m4).(x.0) = f1.(x.0) by FUNCT_7:32,ZF_LANG1:76;
A16:  f1/(x.4,m4).(x.4) = m4 by FUNCT_7:128;
A17:  f1.(x.0) = m by FUNCT_7:128;
A18:  v/(x.3,m3)/(x.0,m).(x.0) = m by FUNCT_7:128;
A19:  M,v/(x.3,m3)/(x.0,m)/(x.4,m4) |= H <=> x.4 '=' x.0 by A12,ZF_LANG1:71;
A20:  now
        assume M,f1/(x.4,m4) |= x.4 '=' x.0;
        then m = m4 by A16,A15,A17,ZF_MODEL:12;
        then M,v/(x.3,m3)/(x.0,m)/(x.4,m4) |= x.4 '=' x.0 by A13,A14,A18,
ZF_MODEL:12;
        then M,v/(x.3,m3)/(x.0,m)/(x.4,m4) |= H by A19,ZF_MODEL:19;
        then M,v/(x.3,m3)/(x.0,m)/(x.4,m4)/(x,v.y) |= H by A5,Th5;
        then M,f1/(x.4,m4) |= H by A7,A8,A9,A11,A1,A2,Th7;
        then M,f1/(x.4,m4)/(y,f1/(x.4,m4).x) |= H by A6,Th9;
        hence M,f1/(x.4,m4) |= F by A5,Th12;
      end;
      now
        assume M,f1/(x.4,m4) |= F;
        then M,f1/(x.4,m4)/(y,f1/(x.4,m4).x) |= H by A5,Th12;
        then M,f1/(x.4,m4) |= H by A6,Th9;
        then M,v/(x.3,m3)/(x.0,m)/(x.4,m4)/(x,v.y) |= H by A7,A8,A9,A11,A1,A2
,Th7;
        then M,v/(x.3,m3)/(x.0,m)/(x.4,m4) |= H by A5,Th5;
        then M,v/(x.3,m3)/(x.0,m)/(x.4,m4) |= x.4 '=' x.0 by A19,ZF_MODEL:19;
        then m = m4 by A13,A14,A18,ZF_MODEL:12;
        hence M,f1/(x.4,m4) |= x.4 '=' x.0 by A16,A15,A17,ZF_MODEL:12;
      end;
      hence M,f1/(x.4,m4) |= F <=> x.4 '=' x.0 by A20,ZF_MODEL:19;
    end;
    then M,f1 |= All(x.4,F <=> x.4 '=' x.0) by ZF_LANG1:71;
    hence M,f/(x.3,m3) |= Ex(x.0,All(x.4,F <=> x.4 '=' x.0)) by ZF_LANG1:73;
  end;
  hence
A21: M,f |= All(x.3,Ex(x.0,All(x.4,F <=> x.4 '=' x.0))) by ZF_LANG1:71;
A22: Free H c= variables_in H by ZF_LANG1:151;
  Free F = Free H by A5,A6,Th2;
  then
A23: not x in Free F by A5,A22;
  let a be Element of M;
  set a9 = def_func'(H,v).a;
  M,v/(x.3,a)/(x.4,a9) |= H by A3,A4,Th10;
  then M,v/(x.3,a)/(x.4,a9)/(x,v.y) |= H by A5,Th5;
  then M,f/(x.3,a)/(x.4,a9) |= H by A8,A9,A2,Th6;
  then M,f/(x.3,a)/(x.4,a9)/(x,f/(x.3,a)/(x.4,a9).y) |= F by A5,Th13;
  then M,f/(x.3,a)/(x.4,a9) |= F by A23,Th9;
  hence def_func'(H,v).a = def_func'(F,f).a by A10,A21,Th10;
end;
