
theorem Th2:
  for M being MetrStruct holds
  ( for a, b being Element of M st dist(a,b) = 0 holds a = b ) iff
  M is discerning
proof
  let M be MetrStruct;
  hereby
    assume
A1: for a, b being Element of M st dist(a,b) = 0 holds a = b;
    the distance of M is discerning
    proof
      let a, b be Element of M;
      assume (the distance of M).(a,b) = 0;
      then dist(a,b) = 0;
      hence thesis by A1;
    end;
    hence M is discerning;
  end;
  assume M is discerning;
  then the distance of M is discerning;
  hence thesis;
end;
