
theorem Th4:
  for A being non diagonal non empty RelStr ex x, y being Element of A st
    x <> y & [x,y] in the InternalRel of A
proof
  let A be non diagonal non empty RelStr;
  now
    assume
A1: for x,y being Element of A st [x,y] in the InternalRel of A holds x = y;
    for a,b being object st [a,b] in the InternalRel of A holds [a,b] in id
    the carrier of A
    proof
      let a,b be object;
      assume
A2:   [a,b] in the InternalRel of A;
      then
A3:   a is Element of A by ZFMISC_1:87;
      b is Element of A by A2,ZFMISC_1:87;
      then a = b by A1,A2,A3;
      hence thesis by A3,RELAT_1:def 10;
    end;
    then the InternalRel of A c= id the carrier of A by RELAT_1:def 3;
    hence thesis by Def1;
  end;
  hence thesis;
end;
