
theorem Th41:
  for P,Q,R,P9,Q9,R9 being
    non point_at_infty Element of ProjectiveSpace TOP-REAL 3
  for h being Element of SubGroupK-isometry
  for N being invertible Matrix of 3,F_Real st h = homography(N) &
  P in BK_model & Q in BK_model & R in absolute &
  P9 = homography(N).P & Q9 = homography(N).Q & R9 = homography(N).R &
  between RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R holds
  between RP3_to_T2 P9,RP3_to_T2 Q9,RP3_to_T2 R9
  proof
    let P,Q,R,P9,Q9,R9 be
      non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    let h be Element of SubGroupK-isometry;
    let N be invertible Matrix of 3,F_Real;
    assume that
A1: h = homography(N) and
A2: P in BK_model and
A3: Q in BK_model and
A4: R in absolute and
A5: P9 = homography(N).P and
A6: Q9 = homography(N).Q and
A7: R9 = homography(N).R and
A8: between RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R;
    consider n11,n12,n13,n21,n22,n23,n31,n32,n33 be Element of F_Real such that
A9: 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
A10: P = Dir u & u`3 = 1 & RP3_to_REAL2 P = |[u`1,u`2]| by Def05;
A11: |[u`1,u`2]| = |[u.1,u`2]| by EUCLID_5:def 1
                .= |[u.1,u.2]| by EUCLID_5:def 2;
    then
A12: P = Dir u & u.3 = 1 & RP3_to_REAL2 P = |[u.1,u.2]| by A10,EUCLID_5:def 3;
    consider v be non zero Element of TOP-REAL 3 such that
A13: R = Dir v & v`3 = 1 & RP3_to_REAL2 R = |[v`1,v`2]| by Def05;
A14: |[v`1,v`2]| = |[v.1,v`2]| by EUCLID_5:def 1
                .= |[v.1,v.2]| by EUCLID_5:def 2;
    then
A15: R = Dir v & v.3 = 1 & RP3_to_REAL2 R = |[v.1,v.2]| by A13,EUCLID_5:def 3;
    consider w be non zero Element of TOP-REAL 3 such that
A16: Q = Dir w & w`3 = 1 & RP3_to_REAL2 Q = |[w`1,w`2]| by Def05;
A17: |[w`1,w`2]| = |[w.1,w`2]| by EUCLID_5:def 1
                .= |[w.1,w.2]| by EUCLID_5:def 2;
    then
A18: Q = Dir w & w.3 = 1 & RP3_to_REAL2 Q = |[w.1,w.2]| by A16,EUCLID_5:def 3;
    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;
    Tn2TR RP3_to_T2 Q in LSeg(Tn2TR RP3_to_T2 P,Tn2TR RP3_to_T2 R)
      by A8,GTARSKI2:20;
    then consider l be Real such that
A19: 0 <= l & l <= 1 and
A20: Tn2TR RP3_to_T2 Q = (1 - l) * Tn2TR RP3_to_T2 P + l * Tn2TR RP3_to_T2 R
      by RLTOPSP1:76;
    |[w.1,w.2]| = |[ (1 - l) * u.1, (1 - l) * u.2 ]| + l * |[v.1,v.2]|
      by A20,A11,A10,A14,A17,A16,A13,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
A21: 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
            .= |[(1 - l) * u.1 + l * v.1, (1 - l) * u.2 + l * v.2,
              (1 - l) * 1 + l * 1]| by A16,A21,EUCLID_5:def 2
            .= |[(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 A12,A15,Th35
            .= (1 - l) * u + l * v by Th35;
      thus nu3 <> 0 by A1,A2,A9,A12,Th20;
      thus nv3 <> 0 by A1,A4,A9,A15,Th22;
      thus
A22:  nw3 <> 0 by A3,A1,A9,A18,Th20;
      thus nw3 = (1 - l) * nu3 + l * nv3 by A21;
      thus (1 - l) * nu3 + l * nv3 <> 0 by A22,A21;
    end;
    then
A23: (1 - r) * |[nu1 / nu3,nu2 / nu3,1]|
      + r * |[nv1 / nv3,nv2 / nv3,1]|
      = |[nw1 / nw3,nw2 / nw3,1]| by Th34;
A24: 0 <= r <= 1
    proof
      now
        thus 0 <= l <= 1 by A19;
        thus 0 < nu3 * nv3
        proof
          reconsider u1 = |[u.1,u.2]|, v1 = |[v.1,v.2]| as
            Element of TOP-REAL 2;
A25:      u1.1 = u1`1 .= u.1 by EUCLID:52;
A26:      u1.2 = u1`2 .= u.2 by EUCLID:52;
A27:      v1.1 = v1`1 .= v.1 by EUCLID:52;
A28:      v1.2 = v1`2 .= v.2 by EUCLID:52;
          reconsider m31 = n31, m32 = n32, m33 = n33 as Element of F_Real;
          now
            reconsider PP = P as Element of BK_model by A2;
            consider u3 be non zero Element of TOP-REAL 3 such that
A29:        Dir u3 = PP & u3.3 = 1 & BK_to_REAL2 PP = |[u3.1,u3.2]|
              by BKMODEL2:def 2;
            u3 = u by A12,A29,Th16;
            hence u1 in inside_of_circle(0,0,1) by A29;
            thus v1 in circle(0,0,1) by A4,A15,BKMODEL1:84;
            thus for w1 be Element of TOP-REAL 2 st
              w1 in closed_inside_of_circle(0,0,1) holds
              m31 * w1.1 + m32 * w1.2 + m33 <> 0 by A1,A9,Th38;
          end;
          hence thesis by A25,A26,A27,A28,Th30;
        end;
      end;
      hence thesis by Th31;
    end;
    now
      thus 0 <= r <= 1 by A24;
      thus Tn2TR RP3_to_T2 Q9 = (1 - r) * Tn2TR RP3_to_T2 P9
                                + r * Tn2TR RP3_to_T2 R9
      proof
        reconsider u2 = |[nu1/nu3,nu2/nu3,1]| as
          non zero Element of TOP-REAL 3;
        reconsider PP9 = P9 as
          non point_at_infty Point of ProjectiveSpace TOP-REAL 3;
        consider u3 be non zero Element of TOP-REAL 3 such that
A30:    PP9 = Dir u3 & u3`3 = 1 & RP3_to_REAL2 PP9 = |[u3`1,u3`2]| by Def05;
        now
          thus Dir u3 = Dir u2 by A1,A2,A9,A12,Th23,A5,A30;
          thus u2.3 = u2`3 by EUCLID_5:def 3
                   .= 1 by EUCLID_5:2;
          hence u2.3 = u3.3 by A30,EUCLID_5:def 3;
        end;
        then u2 = u3 by BKMODEL1:43;
        then
A31:    RP3_to_REAL2 P9 = |[nu1/nu3,u2`2]| by A30,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
A32:    Dir v3= R9 & v3`3 = 1 & RP3_to_REAL2 R9 = |[v3`1,v3`2]| by Def05;
        now
          thus Dir v3 = Dir v2 by A15,A1,A4,A9,Th24,A7,A32;
          thus v2.3 = v2`3 by EUCLID_5:def 3
                   .= 1 by EUCLID_5:2;
          hence v2.3 = v3.3 by A32,EUCLID_5:def 3;
        end;
        then v2 = v3 by BKMODEL1:43;
        then
A33:    RP3_to_REAL2 R9 = |[nv1/nv3,v2`2]| by A32,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
A34:    Dir w3= Q9 & w3`3 = 1 & RP3_to_REAL2 Q9 = |[w3`1,w3`2]| by Def05;
        now
          thus Dir w3 = Dir w2 by A3,A1,A9,A18,Th23,A6,A34;
          thus w2.3 = w2`3 by EUCLID_5:def 3
                   .= 1 by EUCLID_5:2;
          hence w2.3 = w3.3 by A34,EUCLID_5:def 3;
        end;
        then w2 = w3 by BKMODEL1:43;
        then
A35:    RP3_to_REAL2 Q9 = |[nw1/nw3,w2`2]| by A34,EUCLID_5:2
                       .= |[nw1/nw3,nw2/nw3]| by EUCLID_5:2;
        RP3_to_REAL2 Q9 = (1 - r) * RP3_to_REAL2 P9 + r * RP3_to_REAL2 R9
        proof
          reconsider a = nu1/nu3, b = nu2/nu3, c = nv1/nv3, d = nv2/nv3,
          e = nw1/nw3, f = nw2/nw3 as Real;
          (1-r) * |[a,b]| + r * |[c,d]| = |[e,f]| by Th39,A23;
          hence thesis by A31,A33,A35;
        end;
        hence thesis;
      end;
    end;
    then Tn2TR RP3_to_T2 Q9 in {(1 - r) * Tn2TR RP3_to_T2 P9
      + r * Tn2TR RP3_to_T2 R9 where r is Real:0 <= r & r <= 1};
    then Tn2TR RP3_to_T2 Q9 in LSeg(Tn2TR RP3_to_T2 P9,Tn2TR RP3_to_T2 R9)
      by RLTOPSP1:def 2;
    hence thesis by GTARSKI2:20;
  end;
