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 Th11:
  a c= b iff epsilon_a c= epsilon_b
  proof
    hereby assume
      a c= b; then
      a = b or a c< b; then
      a = b or epsilon_a in epsilon_b by ORDINAL1:11,ORDINAL5:44;
      hence epsilon_a c= epsilon_b by ORDINAL1:def 2;
    end;
    assume
A1: epsilon_a c= epsilon_b;
    assume a c/= b; then
    epsilon_b in epsilon_a by ORDINAL1:16,ORDINAL5:44; then
    epsilon_b in epsilon_b by A1;
    hence thesis;
  end;
