
theorem Th1:
  for M being MetrStruct holds ( for a being Element of M holds
  dist(a,a) = 0 ) iff M is Reflexive
proof
  let M be MetrStruct;
  hereby
    assume
A1: for a being Element of M holds dist(a,a) = 0;
    the distance of M is Reflexive
    proof
      let a be Element of M;
      (the distance of M).(a,a) = dist(a,a) .= 0 by A1;
      hence thesis;
    end;
    hence M is Reflexive;
  end;
  assume M is Reflexive;
  then the distance of M is Reflexive;
  hence thesis;
end;
