
theorem ThConv3:
  for A,B being POINT of TarskiEuclid2Space holds
    dist(A,B) = |. Tn2TR A - Tn2TR B .| & dist(A,B) = |. Tn2R A - Tn2R B .|
  proof
    let A,B be POINT of TarskiEuclid2Space;
    the MetrStruct of TarskiEuclid2Space = the MetrStruct of Euclid 2
      by GTARSKI1:def 13;
    then dist(A,B) = (Pitag_dist 2).(Tn2TR A, Tn2TR B) by METRIC_1:def 1
                  .= |. Tn2R A - Tn2R B .| by EUCLID:def 6;
    hence thesis;
end;
