
theorem ThConv4:
  for a,b,c,d being POINT of TarskiEuclid2Space holds
    |. Tn2TR a - Tn2TR b .| = |. Tn2TR c - Tn2TR d .| iff a,b equiv c,d
  proof
    let a,b,c,d be POINT of TarskiEuclid2Space;
A1: dist(a,b) = |. Tn2TR a - Tn2TR b .| &
      dist(c,d) = |. Tn2TR c - Tn2TR d .| by ThConv3;
    thus |. Tn2TR a - Tn2TR b .|= |. Tn2TR c - Tn2TR d .| implies a,b equiv c,d
      by A1,GTARSKI1:def 15;
    assume a,b equiv c,d;
    hence thesis by A1,GTARSKI1:def 15;
  end;
