reserve n,i,k,m for Nat;

theorem Th3:
  for x1,x2,x3 being Element of REAL n holds
  |.x1 - x2.|- |.x2 - x3.| <= |.x1 - x3.|
proof
  let x1,x2,x3 be Element of REAL n;
  |.x1 - x2.|<= |.x1 - x3.|+|.x3 - x2.| by EUCLID:19;
  then |.x1 - x2.|<= |.x1 - x3.|+|.x2 - x3.| by EUCLID:18;
  then |.x1 - x2.|- |.x2 - x3.|<= |.x1 - x3.|+|.x2 - x3.|- |.x2 - x3.|
  by XREAL_1:9;
  hence thesis;
end;
