
theorem Th05:
  for P,Q,R being Point of BK-model-Plane
  for P2,Q2,R2 being POINT of TarskiEuclid2Space st
  P2 = BK_to_T2 P & Q2 = BK_to_T2 Q & R2 = BK_to_T2 R
  holds between P,Q,R iff between P2,Q2,R2
  proof
    let P,Q,R be Point of BK-model-Plane;
    let P2,Q2,R2 be POINT of TarskiEuclid2Space;
    assume that
A1: P2 = BK_to_T2 P and
A2: Q2 = BK_to_T2 Q and
A3: R2 = BK_to_T2 R;
    reconsider p = P,q = Q, r = R as Element of BK_model;
A4: Tn2TR BK_to_T2 P = BK_to_REAL2 p &
      Tn2TR BK_to_T2 Q = BK_to_REAL2 q &
      Tn2TR BK_to_T2 R = BK_to_REAL2 r by Th04;
    hereby
      assume between P,Q,R;
      then BK_to_REAL2 q in LSeg(BK_to_REAL2 p,BK_to_REAL2 r)
        by BKMODEL3:def 7;
      hence between P2,Q2,R2 by A4,A1,A2,A3,GTARSKI2:20;
    end;
    assume between P2,Q2,R2;
    then Tn2TR BK_to_T2 Q in LSeg(Tn2TR BK_to_T2 P,Tn2TR BK_to_T2 R)
      by A1,A2,A3,GTARSKI2:20;
    hence between P,Q,R by A4,BKMODEL3:def 7;
  end;
