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 Th3:
  for x,y,f holds f.x in f.y iff f in St(x 'in' y,E)
proof
  let x,y,f;
A1: x 'in' y is being_membership;
  then
A2: x 'in' y = (Var1(x 'in' y)) 'in' Var2(x 'in' y) by ZF_LANG:37;
  then
A3: x = Var1(x 'in' y) by ZF_LANG:2;
A4: y = Var2(x 'in' y) by A2,ZF_LANG:2;
A5: St(x 'in' y,E) = { v1 : for f st f = v1 holds f.(Var1(x 'in' y)) in f.(
  Var2(x 'in' y)) } by A1,Lm3;
  thus f.x in f.y implies f in St(x 'in' y,E)
  proof
    reconsider v = f as Element of VAL E by FUNCT_2:8;
    assume f.x in f.y;
    then
    for f st f = v holds f.(Var1(x 'in' y)) in f.(Var2(x 'in' y)) by A2,A3,
ZF_LANG:2;
    hence thesis by A5;
  end;
  assume f in St(x 'in' y,E);
  then
  ex v1 st f = v1 & for f st f = v1 holds f.(Var1(x 'in' y)) in f.(Var2(x
  'in' y)) by A5;
  hence thesis by A3,A4;
end;
