reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;

theorem
  for X,Y being ordinal-membered set
  for f being Function st f is_isomorphism_of RelIncl X, RelIncl Y
  for x,y st x in X & y in X holds x in y iff f.x in f.y
  proof
    let X,Y be ordinal-membered set;
    let f be Function; assume
A1: f is_isomorphism_of RelIncl X, RelIncl Y;
    let x,y;
    assume
A2: x in X & y in X;
    field RelIncl X = X & field RelIncl Y = Y by WELLORD2:def 1; then
    dom f = X & rng f = Y by A1; then
    f.x in Y & f.y in Y by A2,FUNCT_1:def 3; then
    reconsider a=f.x,b=f.y,x,y as Ordinal by A2;
    y c= x iff b c= a by A1,A2,Th6;
    hence thesis by Th4;
  end;
