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 Th4:
  RelIncl A,RelIncl B are_isomorphic implies A = B
proof
A1: field RelIncl A = A by Def1;
  assume
A2: RelIncl A,RelIncl B are_isomorphic;
A3: now
A4: field RelIncl B = B by Def1;
    assume
A5: A in B;
    then A = (RelIncl B)-Seg(A) by Th3;
    then RelIncl A = (RelIncl B) |_2 ((RelIncl B)-Seg(A)) by A4,Th1,WELLORD1:9;
    hence contradiction by A2,A5,A4,WELLORD1:40,46;
  end;
  assume A <> B;
  then
A6: A in B or B in A by ORDINAL1:14;
  then B = (RelIncl A)-Seg(B) by A3,Th3;
  then RelIncl B = (RelIncl A) |_2 ((RelIncl A)-Seg(B)) by A1,Th1,WELLORD1:9;
  hence contradiction by A2,A6,A3,A1,WELLORD1:46;
end;
