reserve a,b for object, I,J for set;

theorem Th4:
  for N being multLoopStr, M1,M2 being strict RelExtension of N
  st for m,n being Element of M1 for x,y being Element of M2 st m = x & n = y
     holds m <= n iff x <= y
  holds M1 = M2
  proof
    let N be multLoopStr;
    let M1,M2 be strict RelExtension of N;
    assume
Z4: for m,n being Element of M1 for x,y being Element of M2 st m = x & n = y
    holds m <= n iff x <= y;
A2: the multLoopStr of M1 = the multLoopStr of N &
    the multLoopStr of M2 = the multLoopStr of N by RE;
    the InternalRel of M1 = the InternalRel of M2
    proof let a,b;
      hereby
        assume
A3:     [a,b] in the InternalRel of M1;
        then reconsider m = a, n = b as Element of M1 by ZFMISC_1:87;
        reconsider x = m, y = n as Element of M2 by A2;
        m <= n by A3,ORDERS_2:def 5;
        then x <= y by Z4;
        hence [a,b] in the InternalRel of M2 by ORDERS_2:def 5;
      end;
      assume
A3:   [a,b] in the InternalRel of M2;
      then reconsider m = a, n = b as Element of M2 by ZFMISC_1:87;
      reconsider x = m, y = n as Element of M1 by A2;
      m <= n by A3,ORDERS_2:def 5;
      then x <= y by Z4;
      hence [a,b] in the InternalRel of M1 by ORDERS_2:def 5;
    end;
    hence M1 = M2 by A2;
  end;
