reserve r for Real;

theorem Th25:
  for M being non empty MetrStruct holds
  ( for a, b being Element of M holds a <> b implies 0 < dist(a,b)) iff
  M is Discerning
proof
  let M be non empty MetrStruct;
  hereby
    assume
A1: for a, b being Element of M st a <> b holds 0 < dist(a,b);
    the distance of M is Discerning
    proof
      let a, b be Element of M;
      assume a <> b;
      then 0 < dist(a,b) by A1;
      hence thesis;
    end;
    hence M is Discerning;
  end;
  assume M is Discerning;
   then the distance of M is Discerning;
  hence thesis;
end;
