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 1 = Infty_dist 1
  proof
    now
      let x,y being set;
      assume x in REAL 1 & y in REAL 1;
      then reconsider x1 = x, y1 = y as Element of REAL 1;
      thus (Pitag_dist 1).(x,y) = |.x1.1-y1.1.| by Th66
                               .= (Infty_dist 1).(x,y) by Th65;
    end;
    hence thesis by BINOP_1:def 21;
  end;
