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 Th11:
  Free H c= {x.3,x.4} & M |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '='
  x.0))) implies def_func'(H,v) = def_func(H,M)
proof
  assume that
A1: Free H c= {x.3,x.4} and
A2: M |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0)));
A3: M,v |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0) )) by A2;
  let a be Element of M;
  set r = def_func'(H,v).a;
A4: v/(x.3,a)/(x.4,r).(x.3) = v/(x.3,a).(x.3) by FUNCT_7:32,ZF_LANG1:76;
  not x.0 in Free H by A1,Lm1,Lm2,TARSKI:def 2;
  then
A5: M,v/(x.3,a)/(x.4,r) |= H by A3,Th10;
A6: v/(x.3,a).(x.3) = a by FUNCT_7:128;
  v/(x.3,a)/(x.4,r).(x.4) = r by FUNCT_7:128;
  hence def_func'(H,v).a = def_func(H,M).a by A1,A2,A5,A4,A6,ZFMODEL1:def 2;
end;
