
theorem LemSym:
  for R being Relation holds
    R is symmetric iff
      for x, y being object st [x,y] in R holds [y,x] in R
  proof
    let R be Relation;
    thus R is symmetric implies
      for x, y being object st [x,y] in R holds [y,x] in R
    proof
      assume
  A0: R is symmetric;
      let x, y be object;
      assume A1: [x, y] in R; then
      x in field R & y in field R by RELAT_1:15;
      hence [y, x] in R by A0, A1, RELAT_2:def 3,def 11;
    end;
    assume
A1: for x, y being object st [x,y] in R holds [y,x] in R;
    set X = field R;
    for x,y being object st
      x in X & y in X & [x,y] in R holds [y,x] in R by A1;
    hence R is symmetric by RELAT_2:def 11,RELAT_2:def 3;
  end;
