reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;
reserve n for Nat,
        S for Subset-Family of REAL;
reserve n       for Nat,
        a,b,c,d for Element of REAL n;
reserve n for non zero Nat;
reserve n     for non zero Nat,
        x,y,z for Element of REAL n;

theorem
  Pitag_dist 2 <> Infty_dist 2
  proof
    set x = |[0,0]|, y = |[1,1]|;
    now
      take x,y;
      x is Element of REAL 2 & y is Element of REAL 2 by  EUCLID:22;
      hence x in REAL 2 & y in REAL 2;
      thus (Pitag_dist 2).(x,y) <> (Infty_dist 2).(x,y)
        by Th63,Th64,SQUARE_1:19;
    end;
    hence thesis;
  end;
