reserve X for set,
        A for Subset of X,
        R,S for Relation of X;

theorem Th4:
  R is_symmetric_in X implies R~ is_symmetric_in X
  proof
    assume
A1: R is_symmetric_in X;
    now
      let x,y be object;
      assume that
A2:   x in X and
A3:   y in X and
A4:   [x,y] in R~;
      [y,x] in R by A4,RELAT_1:def 7;
      then [x,y] in R by A2,A3,A1,RELAT_2:def 3;
      hence [y,x] in R~ by RELAT_1:def 7;
    end;
    hence thesis by RELAT_2:def 3;
  end;
