
theorem ThConv:
  for n being Nat
  for p,q being Element of TarskiEuclidSpace n,
      p1, q1 being Element of Euclid n st p = p1 & q = q1 holds
    dist(p,q) = dist(p1, q1)
  proof
    let n be Nat;
    let p,q be Element of TarskiEuclidSpace n;
    let p1,q1 be Element of Euclid n;
    assume
A1: p = p1 & q = q1;
    thus dist(p,q) = (the distance of the MetrStruct of
      TarskiEuclidSpace n).(p,q) by METRIC_1:def 1
                  .= (the distance of the MetrStruct of Euclid n).(p,q)
      by GTARSKI1:def 13
                  .= dist(p1,q1) by METRIC_1:def 1,A1;
  end;
