
theorem Th64:
  for a, q, b, c being POINT of BK-model-Plane holds
  ex x being POINT of BK-model-Plane st between q,a,x & a,x equiv b,c
  proof
    let A,Q,B,C be POINT of BK-model-Plane;
    consider a be Element of BK_model such that
A1: A = a and
    BK_to_T2 A = BK_to_REAL2 a by Def01;
    consider q be Element of BK_model such that
A2: Q = q and
    BK_to_T2 Q = BK_to_REAL2 q by Def01;
    consider b be Element of BK_model such that
A3: B = b and
    BK_to_T2 B = BK_to_REAL2 b by Def01;
    consider c be Element of BK_model such that
A4: C = c and
    BK_to_T2 C = BK_to_REAL2 c by Def01;
    per cases;
    suppose b <> c;
A5:   for q1,a1,b1,c1 being POINT of BK-model-Plane st q1 <> a1 holds
      ex x being POINT of BK-model-Plane st between q1,a1,x & a1,x equiv b1,c1
      proof
        let q1,a1,b1,c1 be POINT of BK-model-Plane;
        assume
A6:     q1 <> a1;
        reconsider Q1 = q1, A1 = a1, B1 = b1, C1 = c1 as Element of BK_model;
        reconsider pQ1=Q1,pA1=A1,pB1=B1,pC1=C1 as
          non point_at_infty Element of ProjectiveSpace TOP-REAL 3 by Th61;
        consider qaR be Element of absolute such that
A7:     (for p,q,r being non point_at_infty Element of
          ProjectiveSpace TOP-REAL 3 st p = q1 & q = a1 & r = qaR holds
          between RP3_to_T2 p,RP3_to_T2 q,RP3_to_T2 r)
          by A6,Th58;
        reconsider pqaR = qaR as
          non point_at_infty Element of ProjectiveSpace TOP-REAL 3 by Th62;
        per cases;
        suppose
A8:       B1 = C1;
          reconsider x = a1 as Element of BK_model;
          reconsider x as POINT of BK-model-Plane;
          take x;
          Tn2TR BK_to_T2 a1 in LSeg(Tn2TR BK_to_T2 q1,Tn2TR BK_to_T2 x)
            by RLTOPSP1:68;
          then between BK_to_T2 q1,BK_to_T2 a1,BK_to_T2 x by GTARSKI2:20;
          hence between q1,a1,x by Th05;
          thus a1,x equiv b1,c1 by A8,Th60;
        end;
        suppose
A9:       B1 <> C1;
          consider bcP be Element of absolute such that
A10:      (for p,q,r being non point_at_infty Element of ProjectiveSpace
            TOP-REAL 3 st p = b1 & q = c1 & r = bcP holds
            between RP3_to_T2 p,RP3_to_T2 q,RP3_to_T2 r) by A9,Th58;
          reconsider pbcP = bcP as non point_at_infty Element of
            ProjectiveSpace TOP-REAL 3 by Th62;
          consider N be invertible Matrix of 3,F_Real such that
A11:      homography(N).:absolute = absolute and
A12:      (homography(N)).B1 = A1 and
A13:      (homography(N)).bcP = qaR by BKMODEL2:56;
          homography(N) in the set of all homography(N) where
          N is invertible Matrix of 3,F_Real;
          then reconsider h = homography(N) as Element of EnsHomography3
            by ANPROJ_9:def 1;
          h is_K-isometry by A11,BKMODEL2:def 6;
          then h in EnsK-isometry by BKMODEL2:def 7;
          then reconsider h = homography(N) as Element of SubGroupK-isometry
            by BKMODEL2:def 8;
          h = homography(N);
          then reconsider x = (homography(N)).C1 as Element of BK_model
            by BKMODEL3:36;
          reconsider x as POINT of BK-model-Plane;
          reconsider px = x as non point_at_infty Element of
          ProjectiveSpace TOP-REAL 3 by Th61;
          take x;
          now
            thus between q1,a1,x
            proof
A14:          between RP3_to_T2 pQ1,RP3_to_T2 pA1,RP3_to_T2 pqaR by A7;
              between RP3_to_T2 pB1,RP3_to_T2 pC1,RP3_to_T2 pbcP &
                h = homography(N) & pB1 in BK_model & pC1 in BK_model &
                pbcP in absolute by A10;
              then
A15:          between RP3_to_T2 pA1,RP3_to_T2 px,RP3_to_T2 pqaR
                by A12,A13,Th41;
              set tq = RP3_to_T2 pQ1, ta = RP3_to_T2 pA1,
                  tx = RP3_to_T2 px, tr = RP3_to_T2 pqaR;
A16:          between tq,ta,tx by A15,A14,GTARSKI3:17;
              now
                consider pp1 be Element of BK_model such that
A17:            q1 = pp1 and
A18:            BK_to_T2 q1 = BK_to_REAL2 pp1 by Def01;
                consider pp2 be Element of BK_model such that
A19:            a1 = pp2 and
A20:            BK_to_T2 a1 = BK_to_REAL2 pp2 by Def01;
                consider pp3 be Element of BK_model such that
A21:            x = pp3 and
A22:            BK_to_T2 x = BK_to_REAL2 pp3 by Def01;
                thus tq = BK_to_T2 q1 by A17,A18,Th63;
                thus ta = BK_to_T2 a1 by A19,A20,Th63;
                thus tx = BK_to_T2 x by Th63,A21,A22;
              end;
              hence thesis by A16,Th05;
            end;
            ex h being Element of SubGroupK-isometry st
            ex N being invertible Matrix of 3,F_Real st
            h = homography(N) & homography(N).a1 = b1 & homography(N).x = c1
            proof
A23:          h = homography(N);
              reconsider M = N~ as invertible Matrix of 3,F_Real;
              reconsider g = homography(M) as Element of SubGroupK-isometry
                by A23,BKMODEL2:47;
              take g;
              ex N being invertible Matrix of 3,F_Real st
                g = homography(N) & homography(N).a1 = b1 &
                homography(N).x = c1
              proof
                take M;
                thus thesis by A12,ANPROJ_9:15;
              end;
              hence thesis;
            end;
            hence a1,x equiv b1,c1 by BKMODEL3:def 8;
          end;
          hence thesis;
        end;
      end;
      q = a implies ex x being POINT of BK-model-Plane st between Q,A,x &
        A,x equiv B,C
      proof
        assume
A24:    q = a;
        consider Q3 be Element of BK_model such that
A25:    a <> Q3 by BKMODEL3:11;
        reconsider q3 = Q3 as Element of BK-model-Plane;
        consider x be POINT of BK-model-Plane such that
        between q3,A,x and
A26:    A,x equiv B,C by A25,A1,A5;
        take x;
        between BK_to_T2 A,BK_to_T2 A,BK_to_T2 x by GTARSKI1:17;
        hence thesis by A26,A1,A24,A2,Th05;
      end;
      hence thesis by A1,A2,A5;
    end;
    suppose
A27:  b = c;
      set X = A;
      take X;
      between BK_to_T2 Q,BK_to_T2 A,BK_to_T2 X by GTARSKI1:14;
      hence between Q,A,X by Th05;
      thus A,A equiv B,C by A27,A3,A4,Th60;
    end;
  end;
