
theorem Th3:
  for M being MetrStruct st
   for a, b being Element of M holds dist(a,b) = dist(b,a) holds
  M is symmetric
proof
  let M be MetrStruct;
    assume
A1: for a, b being Element of M holds dist(a,b) = dist(b,a);
    the distance of M is symmetric
    proof
      let a, b be Element of M;
      thus (the distance of M).(a,b) = dist(a,b) .= dist(b,a) by A1
        .= (the distance of M).(b,a);
    end;
    hence M is symmetric;
end;
