reserve X,Y,Z for set,
  a,b,c,d,x,y,z,u for object,
  R for Relation,
  A,B,C for Ordinal;
reserve H for Function;

theorem
  X c= A implies order_type_of RelIncl X c= A
proof
  assume
A1: X c= A;
  then
A2: (RelIncl A) |_2 X = RelIncl X by Th1;
A3: RelIncl X is well-ordering by A1,Th2;
A4: now
    assume RelIncl A,(RelIncl A) |_2 X are_isomorphic;
    then RelIncl X,RelIncl A are_isomorphic by A2,WELLORD1:40;
    hence thesis by A3,Def2;
  end;
A5: now
    given a being object such that
A6: a in A and
A7: (RelIncl A) |_2 ((RelIncl A)-Seg(a)),(RelIncl A) |_2 X are_isomorphic;
    reconsider a as Ordinal by A6;
A8: (RelIncl A)-Seg(a) = a by A6,Th3;
A9: a c= A by A6,ORDINAL1:def 2;
    then (RelIncl A) |_2 a = RelIncl a by Th1;
    then RelIncl X,RelIncl a are_isomorphic by A2,A7,A8,WELLORD1:40;
    hence thesis by A3,A9,Def2;
  end;
  field RelIncl A = A by Def1;
  hence thesis by A1,A4,A5,WELLORD1:53;
end;
