reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem
  a in b & b in first_epsilon_greater_than a implies
  first_epsilon_greater_than b = first_epsilon_greater_than a
  proof assume
A1: a in b & b in first_epsilon_greater_than a;
    now let c be epsilon Ordinal;
      assume b in c; then
      a in c by A1,ORDINAL1:10;
      hence first_epsilon_greater_than a c= c by Def6;
    end;
    hence first_epsilon_greater_than b = first_epsilon_greater_than a
    by A1,Def6;
  end;
