
theorem
  for R being non empty Poset, s1,s2,s3 being Element of R holds s1 ~=
  s2 & s2 ~= s3 implies s1 ~= s3
proof
  let R be non empty Poset;
  let s1,s2,s3 be Element of R;
  set PR = Path_Rel R;
  field PR = the carrier of R by ORDERS_1:12;
  then
A1: PR is_transitive_in the carrier of R by RELAT_2:def 16;
  assume s1 ~= s2 & s2 ~= s3;
  then [s1,s2] in PR & [s2,s3] in PR;
  then [s1,s3] in PR by A1,RELAT_2:def 8;
  hence thesis;
end;
