reserve X for set, a,b,c,x,y,z for object;
reserve P,R for Relation;

theorem Th6:
  R is symmetric iff R = R~
proof
  hereby
    assume R is symmetric;
    then
A1: R is_symmetric_in field R;
    now
      let a,b be object;
A2:   now
        assume [a,b] in R~;
        then
A3:     [b,a] in R by RELAT_1:def 7;
        then a in field R & b in field R by RELAT_1:15;
        hence [a,b] in R by A1,A3;
      end;
      now
        assume
A4:     [a,b] in R;
        then a in field R & b in field R by RELAT_1:15;
        then [b,a] in R by A1,A4;
        hence [a,b] in R~ by RELAT_1:def 7;
      end;
      hence [a,b] in R iff [a,b] in R~ by A2;
    end;
    hence R = R~;
  end;
  assume R = R~;
  then for a,b holds a in field R & b in field R & [a,b] in R implies [b,a]
  in R by RELAT_1:def 7;
  hence thesis by Def3;
end;
