
theorem ThConv2:
  for p,q being POINT of TarskiEuclid2Space holds
    dist(p,q) =
      sqrt(((Tn2TR p)`1-(Tn2TR q)`1)^2 + (((Tn2TR p)`2-(Tn2TR q)`2)^2))
  proof
    let p,q be POINT of TarskiEuclid2Space;
A1: the MetrStruct of TarskiEuclid2Space = the MetrStruct of Euclid 2
      by GTARSKI1:def 13;
    (Pitag_dist 2).(Tn2E p, Tn2E q) =
      sqrt(( (Tn2TR p)`1 - (Tn2TR q)`1)^2 + ( (Tn2TR p)`2
      - (Tn2TR q)`2)^2) by TOPREAL3:7;
    hence thesis by A1,METRIC_1:def 1;
  end;
