 reserve x,y,z for object, X for set,
         i,k,n,m for Nat,
         R for Relation,
         P for finite Relation,
         p,q for FinSequence;

theorem Th2:
  R is symmetric implies R.:X = R"X
proof
  assume A1:R is symmetric;
  hereby
    let y be object;
    assume y in R.:X;
    then consider z being object such that
          A2: [z,y] in R
      and A3: z in X by RELAT_1:def 13;
    [y,z] in R by A2,A1,Lm3;
    hence y in R"X by A3,RELAT_1:def 14;
  end;
  let y be object;
  assume y in R"X;
  then consider z being object such that
        A4: [y,z] in R
    and A5: z in X by RELAT_1:def 14;
  [z,y] in R by A4,A1,Lm3;
  hence y in R.:X by A5,RELAT_1:def 13;
end;
