
theorem Th41:
  for A being Preorder holds EqRelOf A = EqRel A
proof
  let A be Preorder;
  for x,y being Element of A holds
    [x,y] in EqRelOf A implies [x,y] in EqRel A
  proof
    let x,y be Element of A;
    assume [x,y] in EqRelOf A;
    then x <= y & y <= x by Def6;
    then [x,y] in the InternalRel of A & [y,x] in the InternalRel of A
      by ORDERS_2:def 5;
    then [x,y] in (the InternalRel of A) & [x,y] in (the InternalRel of A)~
      by RELAT_1:def 7;
    then [x,y] in (the InternalRel of A) /\ (the InternalRel of A)~
      by XBOOLE_0:def 4;
    hence thesis by DICKSON:def 4;
  end;
  hence EqRelOf A c= EqRel A by RELSET_1:def 1;
  for x,y being Element of A holds
    [x,y] in EqRel A implies [x,y] in EqRelOf A
  proof
    let x,y be Element of A;
    assume [x,y] in EqRel A;
    then [x,y] in (the InternalRel of A) /\ (the InternalRel of A)~
      by DICKSON:def 4;
    then [x,y] in the InternalRel of A & [x,y] in (the InternalRel of A)~
      by XBOOLE_0:def 4;
    then [x,y] in the InternalRel of A & [y,x] in the InternalRel of A
      by RELAT_1:def 7;
    then x <= y & y <= x by ORDERS_2:def 5;
    hence thesis by Def6;
  end;
  hence EqRel A c= EqRelOf A by RELSET_1:def 1;
end;
