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 Th10:
  not x.0 in Free H & M,v |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '='
x.0))) implies for m1,m2 holds def_func'(H,v).m1 = m2 iff M,v/(x.3,m1)/(x.4,m2)
  |= H
proof
  assume that
A1: not x.0 in Free H and
A2: M,v |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0)));
  let m1,m2;
A3: v/(x.3,m1).(x.3) = m1 by FUNCT_7:128;
A4: now
    let y;
    assume
A5: v/(x.3,m1)/(x.4,m2).y <> v.y;
    assume that
    x.0 <> y and
A6: x.3 <> y and
A7: x.4 <> y;
    v/(x.3,m1)/(x.4,m2).y = v/(x.3,m1).y by A7,FUNCT_7:32;
    hence contradiction by A5,A6,FUNCT_7:32;
  end;
A8: v/(x.3,m1)/(x.4,m2).(x.3) = v/(x.3,m1).(x.3) by FUNCT_7:32,ZF_LANG1:76;
  v/(x.3,m1)/(x.4,m2).(x.4) = m2 by FUNCT_7:128;
  hence thesis by A1,A2,A3,A8,A4,ZFMODEL1:def 1;
end;
