
theorem
  for X being set, R being Relation of X st R is symmetric holds
    R` is symmetric
  proof
    let X be set, R be Relation of X;
    assume
A1: R is symmetric;
    for x, y being object st [x,y] in R` holds [y,x] in R`
    proof
      let x,y be object;
      assume
Z0:   [x,y] in R`; then
xx:   x in field R` & y in field R` by RELAT_1:15;
      R /\ R` = {} by XBOOLE_0:def 7,SUBSET_1:23; then
Z1:   not ([x,y] in R & [x,y] in R`) by XBOOLE_0:def 4;
      assume not [y,x] in R`; then
      [y,x] in R by Lemma12b,xx;
      hence contradiction by Z0,Z1,LemSym,A1;
    end;
    hence R` is symmetric by LemSym;
  end;
