reserve F,H,H9 for ZF-formula,
  x,y,z,t for Variable,
  a,b,c,d,A,X for set;
reserve E for non empty set,
  f,g,h for Function of VAR,E,
  v1,v2,v3,v4,v5,u5 for Element of VAL E;

theorem Th6:
  for x,H,f holds ( f in St(H,E) & for g st for y st g.y <> f.y
  holds x = y holds g in St(H,E) ) iff f in St(All(x,H),E)
proof
  let x,H,f;
A1: All(x,H) is universal;
  then
A2: St(All(x,H),E) = { v5 : for X,f st X = St(the_scope_of All(x,H),E) & f =
  v5 holds f in X & for g st for y st g.y <> f.y holds bound_in All(x,H) = y
  holds g in X } by Lm3;
A3: All(x,H) = All(bound_in All(x,H),the_scope_of All(x,H)) by A1,ZF_LANG:44;
  then
A4: x = bound_in All(x,H) by ZF_LANG:3;
A5: H = the_scope_of All(x,H) by A3,ZF_LANG:3;
  thus ( f in St(H,E) & for g st for y st g.y <> f.y holds x = y holds g in St
  (H,E) ) implies f in St(All(x,H),E)
  proof
    reconsider v = f as Element of VAL E by FUNCT_2:8;
    assume
    f in St(H,E) & for g st for y st g.y <> f.y holds x = y holds g in St(H,E);
    then
    for X,h holds X = St(the_scope_of All(x,H),E) & h = v implies h in X &
for g holds ( for y st g.y <> h.y holds bound_in All(x,H) = y ) implies g in X
    by A4,A5;
    hence thesis by A2;
  end;
  assume f in St(All(x,H),E);
  then
A6: ex v5 st f = v5 & for X,f st X = St(the_scope_of All(x,H),E) & f = v5
holds f in X & for g st for y st g.y <> f.y holds bound_in All(x,H) = y holds g
  in X by A2;
  hence f in St(H,E) by A5;
  let g;
  assume for y st g.y <> f.y holds x = y;
  hence thesis by A4,A5,A6;
end;
