
theorem ThEquiv:
  for n being Nat
  for p,q being POINT of TarskiEuclidSpace n,
      p1,q1 being Element of TOP-REAL n st p = p1 & q = q1 holds
    dist(p,q) = (Pitag_dist n).(p1, q1) & dist(p,q) = |. p1 - q1 .|
  proof
    let n be Nat;
    let p,q be POINT of TarskiEuclidSpace n,
        p1,q1 being Element of TOP-REAL n;
    assume
A0: p = p1 & q = q1;
A1: the MetrStruct of TarskiEuclidSpace n = the MetrStruct of Euclid n
      by GTARSKI1:def 13; then
    (Pitag_dist n).(p1, q1) = |. p1 - q1 .| by A0,EUCLID:def 6;
    hence thesis by A1,METRIC_1:def 1,A0;
  end;
