
theorem ::Exercise 4.27:
  for R being non empty Poset, x being Element of R
  holds Class(EqRel R, x) = {x}
proof
  let R be non empty Poset;
  set IR = the InternalRel of R, CR = the carrier of R;
  let x be Element of CR;
A1: R is quasi_ordered;
A2: IR is_antisymmetric_in CR by ORDERS_2:def 4;
  now
    let z be object;
    hereby
      assume z in Class(EqRel R, x);
      then [z,x] in EqRel R by EQREL_1:19;
      then
A3:   [z,x] in IR /\ IR~ by A1,Def4;
      then
A4:   [z,x] in IR by XBOOLE_0:def 4;
      [z,x] in IR~ by A3,XBOOLE_0:def 4;
      then
A5:   [x,z] in IR by RELAT_1:def 7;
      z in dom IR by A4,XTUPLE_0:def 12;
      then z = x by A2,A4,A5;
      hence z in {x} by TARSKI:def 1;
    end;
    assume z in {x};
    then z = x by TARSKI:def 1;
    hence z in Class(EqRel R, x) by EQREL_1:20;
  end;
  hence thesis by TARSKI:2;
end;
