
theorem
  for P,Q,R,P9,Q9,R9 being POINT of BK-model-Plane
  for p,q,r,p9,q9,r9 being Element of BK_model
  for h being Element of SubGroupK-isometry
  for N being invertible Matrix of 3,F_Real st h = homography(N) &
  between P,Q,R & P = p & Q = q & R = r & p9 = homography(N).p &
  q9 = homography(N).q & r9 = homography(N).r &
  P9 = p9 & Q9 = q9 & R9 = r9 holds between P9,Q9,R9
  proof
    let P,Q,R,P9,Q9,R9 be POINT of BK-model-Plane;
    let p,q,r,p9,q9,r9 be Element of BK_model;
    let h be Element of SubGroupK-isometry;
    let N be invertible Matrix of 3,F_Real;
    assume that
A1: h = homography(N) and
A2: between P,Q,R and
A3: P = p and
A4: Q = q and
A5: R = r and
A6: p9 = homography(N).p and
A7: q9 = homography(N).q and
A8: r9 = homography(N).r and
A9: P9 = p9 and
A10: Q9 = q9 and
A11: R9 = r9;
    consider n11,n12,n13,n21,n22,n23,n31,n32,n33 be Element of F_Real such that
A12: N = <* <* n11,n12,n13 *>,
            <* n21,n22,n23 *>,
            <* n31,n32,n33 *> *> by PASCAL:3;
    consider u be non zero Element of TOP-REAL 3 such that
A13: Dir u = p & u.3 = 1 & BK_to_REAL2 p = |[u.1,u.2]| by BKMODEL2:def 2;
    consider v be non zero Element of TOP-REAL 3 such that
A14: Dir v = r & v.3 = 1 & BK_to_REAL2 r = |[v.1,v.2]| by BKMODEL2:def 2;
    consider w be non zero Element of TOP-REAL 3 such that
A15: Dir w = q & w.3 = 1 & BK_to_REAL2 q = |[w.1,w.2]| by BKMODEL2:def 2;
    reconsider nu1 = n11 * u.1 + n12 * u.2 + n13,
      nu2 = n21 * u.1 + n22 * u.2 + n23,
      nu3 = n31 * u.1 + n32 * u.2 + n33,
      nv1 = n11 * v.1 + n12 * v.2 + n13,
      nv2 = n21 * v.1 + n22 * v.2 + n23,
      nv3 = n31 * v.1 + n32 * v.2 + n33,
      nw1 = n11 * w.1 + n12 * w.2 + n13,
      nw2 = n21 * w.1 + n22 * w.2 + n23,
      nw3 = n31 * w.1 + n32 * w.2 + n33 as Real;
A16: BK_to_T2 P = BK_to_REAL2 p & Tn2TR BK_to_T2 P = BK_to_REAL2 p &
      BK_to_T2 Q = BK_to_REAL2 q & Tn2TR BK_to_T2 Q = BK_to_REAL2 q &
      BK_to_T2 R = BK_to_REAL2 r & Tn2TR BK_to_T2 R = BK_to_REAL2 r
      by A3,A4,A5,Th04;
    then Tn2TR BK_to_T2 Q in LSeg(Tn2TR BK_to_T2 P,Tn2TR BK_to_T2 R)
      by A2,A3,A4,A5,BKMODEL3:def 7;
    then consider l be Real such that
A17: 0 <= l & l <= 1 and
A18: Tn2TR BK_to_T2 Q = (1 - l) * Tn2TR BK_to_T2 P + l * Tn2TR BK_to_T2 R
      by RLTOPSP1:76;
    |[w.1,w.2]| = |[ (1 - l) * u.1, (1 - l) * u.2 ]| + l * |[v.1,v.2]|
      by A18,A16,A13,A14,A15,EUCLID:58
               .= |[ (1 - l) * u.1, (1 - l) * u.2 ]| + |[l * v.1,l * v.2]|
                 by EUCLID:58
               .= |[ (1 - l) * u.1 + l * v.1, (1 - l) * u.2 + l * v.2]|
                 by EUCLID:56;
    then
A19: w.1 = (1 - l) * u.1 + l * v.1 &
    w.2 = (1 - l) * u.2 + l * v.2 by FINSEQ_1:77;
    set r = (l * nv3) / ((1 - l) * nu3 + l * nv3);
    now
      thus w = |[w`1,w`2,w`3]| by EUCLID_5:3
            .= |[w.1,w`2,w`3]| by EUCLID_5:def 1
            .= |[w.1,w.2,w`3]| by EUCLID_5:def 2
            .= |[(1 - l) * u.1 + l * v.1, (1 - l) * u.2 + l * v.2,
              (1 - l) * 1 + l * 1]| by A15,A19,EUCLID_5:def 3
            .= |[(1 -l) * u.1,(1 - l) * u.2,(1 - l) * 1]| +
              |[ l * v.1,l * v.2, l * 1]| by EUCLID_5:6
            .= (1 - l) * |[u.1,u.2,1]| +
              |[ l * v.1,l * v.2, l * 1]| by EUCLID_5:8
            .= (1 - l) * |[u.1,u.2,1]| +
              l * |[v.1,v.2, 1]| by EUCLID_5:8
            .= (1 - l) * u + l * |[v.1,v.2,v.3]| by A13,A14,Th35
            .= (1 - l) * u + l * v by Th35;
      thus nu3 <> 0 by A1,A12,A13,Th20;
      thus nv3 <> 0 by A1,A12,A14,Th20;
      thus
A20:  nw3 <> 0 by A1,A12,A15,Th20;
      thus nw3 = (1 - l) * nu3 + l * nv3 by A19;
      thus (1 - l) * nu3 + l * nv3 <> 0 by A19,A20;
    end;
    then
A21: (1 - r) * |[nu1 / nu3,nu2 / nu3,1]| + r * |[nv1 / nv3,nv2 / nv3,1]|
      = |[nw1 / nw3,nw2 / nw3,1]| by Th34;
A22: 0 <= r <= 1
    proof
      now
        thus 0 <= l <= 1 by A17;
        thus 0 < nu3 * nv3
        proof
          reconsider u1 = |[u.1,u.2]|, v1 = |[v.1,v.2]| as
            Element of TOP-REAL 2;
A23:      u1.1 = u1`1 .= u.1 by EUCLID:52;
A24:      u1.2 = u1`2 .= u.2 by EUCLID:52;
A25:      v1.1 = v1`1 .= v.1 by EUCLID:52;
A26:      v1.2 = v1`2 .= v.2 by EUCLID:52;
          reconsider m31 = n31, m32 = n32, m33 = n33 as Element of F_Real;
          u1 in inside_of_circle(0,0,1) &
            v1 in inside_of_circle(0,0,1) &
            for w1 be Element of TOP-REAL 2 st
              w1 in inside_of_circle(0,0,1) holds
              m31 * w1.1 + m32 * w1.2 + m33 <> 0 by A13,A14,A1,A12,Th36;
          hence thesis by A23,A24,A25,A26,Th29;
        end;
      end;
      hence thesis by Th31;
    end;
A27: BK_to_T2 P9 = BK_to_REAL2 p9 &
      Tn2TR BK_to_T2 P9 = BK_to_REAL2 p9 &
      BK_to_T2 Q9 = BK_to_REAL2 q9 &
      Tn2TR BK_to_T2 Q9 = BK_to_REAL2 q9 &
      BK_to_T2 R9 = BK_to_REAL2 r9 &
      Tn2TR BK_to_T2 R9 = BK_to_REAL2 r9
      by A9,A10,A11,Th04;
    now
      thus 0 <= r <= 1 by A22;
      thus Tn2TR BK_to_T2 Q9 = (1 - r) * Tn2TR BK_to_T2 P9
                                + r * Tn2TR BK_to_T2 R9
      proof
        reconsider u2 = |[nu1/nu3,nu2/nu3,1]|
          as non zero Element of TOP-REAL 3;
        consider u3 be non zero Element of TOP-REAL 3 such that
A28:    Dir u3 = p9 & u3.3 = 1 & BK_to_REAL2 p9 = |[u3.1,u3.2]|
          by BKMODEL2:def 2;
        now
          thus Dir u3 = Dir u2 by A28,A6,A1,A12,A13,Th23;
          thus u2.3 = u2`3 by EUCLID_5:def 3
                   .= 1 by EUCLID_5:2;
          hence u2.3 = u3.3 by A28;
        end;
        then u2 = u3 by BKMODEL1:43;
        then
A29:    BK_to_REAL2 p9 = |[u2`1,u2.2]| by A28,EUCLID_5:def 1
                      .= |[u2`1,u2`2]| by EUCLID_5:def 2
                      .= |[nu1/nu3,u2`2]| by EUCLID_5:2
                      .= |[nu1/nu3,nu2/nu3]| by EUCLID_5:2;
        reconsider v2 = |[nv1/nv3,nv2/nv3,1]| as
          non zero Element of TOP-REAL 3;
        consider v3 be non zero Element of TOP-REAL 3 such that
A30:    Dir v3 = r9 & v3.3 = 1 & BK_to_REAL2 r9 = |[v3.1,v3.2]|
          by BKMODEL2:def 2;
        now
          thus Dir v3 = Dir v2 by A30,A1,A12,A14,Th23,A8;
          thus v2.3 = v2`3 by EUCLID_5:def 3
                   .= 1 by EUCLID_5:2;
          hence v2.3 = v3.3 by A30;
        end;
        then v2 = v3 by BKMODEL1:43;
        then
A31:    BK_to_REAL2 r9 = |[v2`1,v2.2]| by A30,EUCLID_5:def 1
                      .= |[v2`1,v2`2]| by EUCLID_5:def 2
                      .= |[nv1/nv3,v2`2]| by EUCLID_5:2
                      .= |[nv1/nv3,nv2/nv3]| by EUCLID_5:2;
        reconsider w2 = |[nw1/nw3,nw2/nw3,1]|
          as non zero Element of TOP-REAL 3;
        consider w3 be non zero Element of TOP-REAL 3 such that
A32:    Dir w3 = q9 & w3.3 = 1 & BK_to_REAL2 q9 = |[w3.1,w3.2]|
          by BKMODEL2:def 2;
        now
          thus Dir w3 = Dir w2 by A32,A1,A12,A15,Th23,A7;
          thus w2.3 = w2`3 by EUCLID_5:def 3
                   .= 1 by EUCLID_5:2;
          hence w2.3 = w3.3 by A32;
        end;
        then w2 = w3 by BKMODEL1:43;
        then BK_to_REAL2 q9 = |[w2`1,w2.2]| by A32,EUCLID_5:def 1
                           .= |[w2`1,w2`2]| by EUCLID_5:def 2
                           .= |[nw1/nw3,w2`2]| by EUCLID_5:2
                           .= |[nw1/nw3,nw2/nw3]| by EUCLID_5:2;
        hence thesis by A27,A29,A31,Th39,A21;
      end;
    end;
    then Tn2TR BK_to_T2 Q9 in {(1 - r) * Tn2TR BK_to_T2 P9
      + r * Tn2TR BK_to_T2 R9 where r is Real:0 <= r & r <= 1};
    then Tn2TR BK_to_T2 Q9 in LSeg(Tn2TR BK_to_T2 P9,Tn2TR BK_to_T2 R9)
      by RLTOPSP1:def 2;
    hence thesis by A27,A9,A10,A11,BKMODEL3:def 7;
  end;
