
theorem Th4:
  for M being MetrStruct holds ( for a, b, c being Element of M
  holds dist(a,c) <= dist(a,b) + dist(b,c) ) iff M is triangle
proof
  let M be MetrStruct;
  hereby
    assume
A1: for a, b, c being Element of M holds dist(a,c) <= dist(a,b) + dist (b,c);
    the distance of M is triangle
    proof
      let a, b, c be Element of M;
A2:   (the distance of M).(b,c) = dist(b,c);
      (the distance of M).(a,b) = dist(a,b) &
      (the distance of M).(a,c) = dist(a,c);
      hence thesis by A1,A2;
    end;
    hence M is triangle;
  end;
  assume
A3: M is triangle;
  let a, b, c be Element of M;
  the distance of M is triangle by A3;
  hence thesis;
end;
