
theorem
  for P,Q being Point of BK-model-Plane st P <> Q holds
  length(P,P,Q) = 0 & length(P,Q,Q) = 1
  proof
    let P,Q be Point of BK-model-Plane;
    assume
A1: P <> Q;
    reconsider P2 = BK_to_T2 P,Q2 = BK_to_T2 Q as Point of TarskiEuclid2Space;
A2: between P,P,Q by Th09;
A3: between P,Q,Q by Th09;
    reconsider r = 0 as Real;
    now
      thus 0 <= r <= 1;
      thus (1 - r ) * Tn2TR BK_to_T2 P + r * Tn2TR BK_to_T2 Q
        = |[ 1 * (Tn2TR BK_to_T2 P)`1, 1 * (Tn2TR BK_to_T2 P)`2 ]|
          + 0 * Tn2TR BK_to_T2 Q by EUCLID:57
       .= |[ (Tn2TR BK_to_T2 P)`1, (Tn2TR BK_to_T2 P)`2 ]|
          + |[ 0 * (Tn2TR BK_to_T2 Q)`1, 0 * (Tn2TR BK_to_T2 Q)`2 ]|
          by EUCLID:57
       .= |[ (Tn2TR BK_to_T2 P)`1 + 0 , (Tn2TR BK_to_T2 P)`2 + 0 ]|
          by EUCLID:56
       .= Tn2TR BK_to_T2 P by EUCLID:53;
    end;
    hence length(P,P,Q) = 0 by A1,A2,Def03;
    reconsider s = 1 as Real;
    now
      thus 0 <= s <= 1;
      thus (1 - s ) * Tn2TR BK_to_T2 P + s * Tn2TR BK_to_T2 Q
        = |[ 1 * (Tn2TR BK_to_T2 Q)`1, 1 * (Tn2TR BK_to_T2 Q)`2 ]|
          + 0 * Tn2TR BK_to_T2 P by EUCLID:57
       .= |[ (Tn2TR BK_to_T2 Q)`1, (Tn2TR BK_to_T2 Q)`2 ]|
          + |[ 0 * (Tn2TR BK_to_T2 P)`1, 0 * (Tn2TR BK_to_T2 P)`2 ]|
          by EUCLID:57
       .= |[ (Tn2TR BK_to_T2 Q)`1 + 0 , (Tn2TR BK_to_T2 Q)`2 + 0 ]|
          by EUCLID:56
       .= Tn2TR BK_to_T2 Q by EUCLID:53;
    end;
    hence length(P,Q,Q) = 1 by A1,A3,Def03;
  end;
