reserve P for Element of BK_model;
reserve N,N1,N2 for invertible Matrix of 3,F_Real;
reserve l,l1,l2 for Element of the Lines of IncProjSp_of real_projective_plane;
reserve P for Point of ProjectiveSpace TOP-REAL 3,
        l for LINE of IncProjSp_of real_projective_plane;

theorem Th29:
  not P in BK_model \/ absolute implies (ex l st P in l &
  l misses absolute)
  proof
    assume
A1: not P in BK_model \/ absolute;
    then
A2: not P in BK_model & not P in absolute by XBOOLE_0:def 3;
    consider u9 be Element of TOP-REAL 3 such that
A3: u9 is not zero and
A4: P = Dir u9 by ANPROJ_1:26;
    per cases;
    suppose
A5:   u9.3 = 0;
      u9.1 <> 0 or u9.2 <> 0
      proof
        assume
A6:     u9.1 = 0 & u9.2 = 0;
        u9 = |[u9`1,u9`2,u9`3]| by EUCLID_5:3
          .= |[0,u9`2,u9`3]| by A6,EUCLID_5:def 1
          .= |[0,0,u9`3]| by A6,EUCLID_5:def 2
          .= 0.TOP-REAL 3 by A5,EUCLID_5:def 3,4;
        hence contradiction by A3;
      end;
      then per cases;
      suppose
A7:     u9.1 <> 0;
        then reconsider v = |[-u9.2,u9.1,0]| as non zero Element of TOP-REAL 3;
        reconsider Q = Dir v as Point of ProjectiveSpace TOP-REAL 3
          by ANPROJ_1:26;
        P <> Q
        proof
          assume P = Q;
          then are_Prop u9,v by A3,A4,ANPROJ_1:22;
          then consider a be Real such that
          a <> 0 and
A9:       u9 = a * v by ANPROJ_1:1;
A10:      |[a * (- u9.2), a * u9.1,a * 0]|
            = a * v by EUCLID_5:8
           .= |[u9`1,u9`2,u9`3]| by A9,EUCLID_5:3;
          now
            thus a * (- u9.2) = |[u9`1,u9`2,u9`3]|`1 by A10,EUCLID_5:2
                             .= u9`1 by EUCLID_5:2
                             .= u9.1 by EUCLID_5:def 1;
            thus a * u9.1 = |[u9`1,u9`2,u9`3]|`2 by A10,EUCLID_5:2
                         .= u9`2 by EUCLID_5:2
                         .= u9.2 by EUCLID_5:def 2;
          end;
          then u9.1 = 0 & u9.2 = 0 by Th10;
          then u9`1 = 0 & u9`2 = 0 & u9`3 = 0 by A5,EUCLID_5:def 1,def 2,def 3;
          hence contradiction by A3,EUCLID_5:3,4;
        end;
        then reconsider l9 = Line(P,Q) as LINE of real_projective_plane
          by COLLSP:def 7;
        reconsider l = l9 as Element of the Lines of
          IncProjSp_of real_projective_plane by INCPROJ:4;
        take l;
        l misses absolute
        proof
          assume not l misses absolute;
          then consider R be object such that
A11:      R in l /\ absolute by XBOOLE_0:7;
A12:      R in l & R in absolute by A11,XBOOLE_0:def 4;
          reconsider R as Element of ProjectiveSpace TOP-REAL 3 by A11;
          consider w be non zero Element of TOP-REAL 3 such that
A13:      w.3 = 1 and
A14:      R = Dir w by A12,Th25;
          reconsider R9 = R as POINT of IncProjSp_of real_projective_plane
            by INCPROJ:3;
          reconsider l2 = l as LINE of IncProjSp_of real_projective_plane;
A15:      w`3 = 1 by A13,EUCLID_5:def 3;
A16:      u9`3 = 0 by A5,EUCLID_5:def 3;
A17:      v`1 = - u9.2 by EUCLID_5:2
             .= -u9`2 by EUCLID_5:def 2;
A18:      v`2 = u9.1 by EUCLID_5:2
             .= u9`1 by EUCLID_5:def 1;
A19:      v`3 = 0 by EUCLID_5:2;
          R9 on l2 by A12,INCPROJ:5;
          then |{u9,v,w}| = 0 by A3,A4,A14,BKMODEL1:77;
          then 0 = u9`1 * v`2 * 1 - u9`3*v`2*w`1 - u9`1*v`3*w`2
                    + u9`2*v`3*w`1 -
                    u9`2*v`1* 1 + u9`3*v`1*w`2 by A15,ANPROJ_8:27
                .= (u9`1)^2 + (u9`2)^2 by A16,A17,A18,A19;
          then u9`1 = 0 & u9`2 = 0;
          hence contradiction by A7,EUCLID_5:def 1;
        end;
        hence thesis by COLLSP:10;
      end;
      suppose
A20:    u9.2 <> 0;
        then reconsider v = |[-u9.2,u9.1,0]| as non zero Element of TOP-REAL 3;
        reconsider Q = Dir v as Point of ProjectiveSpace TOP-REAL 3
          by ANPROJ_1:26;
        P <> Q
        proof
          assume P = Q;
          then are_Prop u9,v by A3,A4,ANPROJ_1:22;
          then consider a be Real such that
          a <> 0 and
A22:      u9 = a * v by ANPROJ_1:1;
A23:      |[a * (- u9.2), a * u9.1,a * 0]|
            = a * v by EUCLID_5:8
           .= |[u9`1,u9`2,u9`3]| by A22,EUCLID_5:3;
          now
            thus a * (- u9.2) = |[u9`1,u9`2,u9`3]|`1 by A23,EUCLID_5:2
                             .= u9`1 by EUCLID_5:2
                             .= u9.1 by EUCLID_5:def 1;
            thus a * u9.1 = |[u9`1,u9`2,u9`3]|`2 by A23,EUCLID_5:2
                         .= u9`2 by EUCLID_5:2
                         .= u9.2 by EUCLID_5:def 2;
          end;
          then u9.1 = 0 & u9.2 = 0 by Th10;
          then u9`1 = 0 & u9`2 = 0 & u9`3 = 0 by A5,EUCLID_5:def 1,def 2,def 3;
          hence contradiction by A3,EUCLID_5:3,4;
        end;
        then reconsider l9 = Line(P,Q) as LINE of real_projective_plane
          by COLLSP:def 7;
        reconsider l = l9 as Element of the Lines of
          IncProjSp_of real_projective_plane by INCPROJ:4;
        take l;
        l misses absolute
        proof
          assume not l misses absolute;
          then consider R be object such that
A24:      R in l /\ absolute by XBOOLE_0:7;
A25:      R in l & R in absolute by A24,XBOOLE_0:def 4;
          reconsider R as Element of ProjectiveSpace TOP-REAL 3 by A24;
          consider w be non zero Element of TOP-REAL 3 such that
A26:      w.3 = 1 and
A27:      R = Dir w by A25,Th25;
          reconsider R9 = R as POINT of IncProjSp_of real_projective_plane
            by INCPROJ:3;
          reconsider l2 = l as LINE of IncProjSp_of real_projective_plane;
A28:      w`3 = 1 by A26,EUCLID_5:def 3;
A29:      u9`3 = 0 by A5,EUCLID_5:def 3;
A30:      v`1 = - u9.2 by EUCLID_5:2
             .= -u9`2 by EUCLID_5:def 2;
A31:      v`2 = u9.1 by EUCLID_5:2
             .= u9`1 by EUCLID_5:def 1;
A32:      v`3 = 0 by EUCLID_5:2;
          R9 on l2 by A25,INCPROJ:5;
          then |{u9,v,w}| = 0 by A3,A4,A27,BKMODEL1:77;
          then
          0 = u9`1 * v`2 * 1 - u9`3*v`2*w`1 - u9`1*v`3*w`2
                 + u9`2*v`3*w`1 -
                 u9`2*v`1* 1 + u9`3*v`1*w`2 by A28,ANPROJ_8:27
           .= (u9`1)^2 + (u9`2)^2 by A29,A30,A31,A32;
          then u9`1 = 0 & u9`2 = 0;
          hence contradiction by A20,EUCLID_5:def 2;
        end;
        hence thesis by COLLSP:10;
      end;
    end;
    suppose
A33:  u9.3 <> 0;
      reconsider u = |[u9.1 / u9.3, u9.2 / u9.3, 1]| as
        non zero Element of TOP-REAL 3;
A34:  u`3 = 1 by EUCLID_5:2;
      then
A35:  u.3 = 1 by EUCLID_5:def 3;
      u9.3 * u = |[u9.1,u9.2,u9.3]| by A33,Th22
              .= |[u9`1,u9.2,u9.3]| by EUCLID_5:def 1
              .= |[u9`1,u9`2,u9.3]| by EUCLID_5:def 2
              .= |[u9`1,u9`2,u9`3]| by EUCLID_5:def 3
              .= u9 by EUCLID_5:3;
      then
A36:  are_Prop u,u9 by A33,ANPROJ_1:1;
      then
A37:  P = Dir u by A3,A4,ANPROJ_1:22;
      u.1 <> 0 or u.2 <> 0
      proof
        assume
A38:    u.1 = 0 & u.2 = 0;
        now
          let v be Element of TOP-REAL 3;
          assume
A39:      v is non zero;
          assume P = Dir v;
          then
A40:      Dir u = Dir v by A36,A3,A4,ANPROJ_1:22;
          now
            thus 0 < (u.3)^2 by A35;
            thus qfconic(1,1,-1,0,0,0,u) = 1 * u.1 * u.1 + 1 * u.2 * u.2
                + (-1) * u.3 * u.3 +
                0 * u.1 * u.2 + 0 * u.1 * u.3 + 0 * u.2 * u.3 by PASCAL:def 1
              .= (-1) * (u.3)^2 by A38;
            end;
            hence qfconic(1,1,-1,0,0,0,v) is negative by A39,A40,BKMODEL1:81;
          end;
          hence contradiction by A2,BKMODEL2:def 1;
        end;
        then per cases;
        suppose
A41:      u.1 <> 0;
          then reconsider u1 = u.1 as non zero Real;
          reconsider u2 = u.2 as Real;
          reconsider v = |[-u2,u1,0]| as non zero Element of TOP-REAL 3;
          reconsider Q = Dir v as Element of ProjectiveSpace TOP-REAL 3
            by ANPROJ_1:26;
          reconsider l9 = Line(P,Q) as LINE of real_projective_plane
            by Th26,A37,COLLSP:def 7;
          reconsider l = l9 as Element of the Lines of
            IncProjSp_of real_projective_plane by INCPROJ:4;
          take l;
          l misses absolute
          proof
            assume not l misses absolute;
            then consider R be object such that
A42:        R in l /\ absolute by XBOOLE_0:7;
A43:        R in l & R in absolute by A42,XBOOLE_0:def 4;
            reconsider R as Element of ProjectiveSpace TOP-REAL 3 by A42;
            consider w be non zero Element of TOP-REAL 3 such that
A44:        w.3 = 1 and
A45:        R = Dir w by A43,Th25;
            |[w.1,w.2]| in circle(0,0,1) by A43,A44,A45,BKMODEL1:84;
            then (w.1)^2 + (w.2)^2 = 1 by BKMODEL1:13;
            then (w`1)^2 + (w.2)^2 = 1 by EUCLID_5:def 1;
            then
A46:        (w`1)^2 + (w`2)^2 = 1 by EUCLID_5:def 2;
            now
              thus u`1 <> 0 by A41,EUCLID_5:def 1;
              thus u`3 = 1 by EUCLID_5:2;
              v`1 = -u2 by EUCLID_5:2;
              hence v`1 = -u`2 by EUCLID_5:def 2;
              v`2 = u.1 by EUCLID_5:2;
              hence v`2 = u`1 by EUCLID_5:def 1;
              thus v`3 = 0 by EUCLID_5:2;
              thus w`3 = 1 by A44,EUCLID_5:def 3;
              reconsider R9 = R as POINT of IncProjSp_of
                real_projective_plane by INCPROJ:3;
              reconsider l2 = l as LINE of IncProjSp_of real_projective_plane;
              R9 on l2 by A43,INCPROJ:5;
              hence |{u,v,w}| = 0 by A37,A45,BKMODEL1:77;
              thus 1 < (u`1)^2 + (u`2)^2
              proof
                assume not 1 < (u`1)^2 + (u`2)^2;
                then Dir |[u`1,u`2,1]| in BK_model \/ absolute by Th28;
                hence contradiction by A37,A1,A34,EUCLID_5:3;
              end;
            end;
            hence contradiction by A46,Th23;
          end;
          hence thesis by COLLSP:10;
        end;
        suppose
A47:      u.2 <> 0;
          then reconsider u2 = u.2 as non zero Real;
          reconsider u1 = u.1 as Real;
          reconsider v = |[-u2,u1,0]| as non zero Element of TOP-REAL 3;
          reconsider Q = Dir v as Element of ProjectiveSpace TOP-REAL 3
            by ANPROJ_1:26;
          reconsider l9 = Line(P,Q) as LINE of real_projective_plane
            by A37,Th26,COLLSP:def 7;
          reconsider l = l9 as Element of the Lines of IncProjSp_of
            real_projective_plane by INCPROJ:4;
          take l;
          l misses absolute
          proof
            assume l meets absolute;
            then consider R be object such that
A48:        R in l /\ absolute by XBOOLE_0:7;
A49:        R in l & R in absolute by A48,XBOOLE_0:def 4;
            reconsider R as Element of ProjectiveSpace TOP-REAL 3 by A48;
            consider w be non zero Element of TOP-REAL 3 such that
A50:        w.3 = 1 and
A51:        R = Dir w by A49,Th25;
            |[w.1,w.2]| in circle(0,0,1) by A49,A50,A51,BKMODEL1:84;
            then (w.1)^2 + (w.2)^2 = 1 by BKMODEL1:13;
            then (w`1)^2 + (w.2)^2 = 1 by EUCLID_5:def 1;
            then
A53:        (w`1)^2 + (w`2)^2 = 1 by EUCLID_5:def 2;
            now
              thus u`2 <> 0 by A47,EUCLID_5:def 2;
              thus u`3 = 1 by EUCLID_5:2;
              v`1 = -u2 by EUCLID_5:2;
              hence v`1 = -u`2 by EUCLID_5:def 2;
              v`2 = u.1 by EUCLID_5:2;
              hence v`2 = u`1 by EUCLID_5:def 1;
              thus v`3 = 0 by EUCLID_5:2;
              thus w`3 = 1 by A50,EUCLID_5:def 3;
              reconsider R9 = R as POINT of IncProjSp_of
                real_projective_plane by INCPROJ:3;
              reconsider l2 = l as LINE of IncProjSp_of real_projective_plane;
              R9 on l2 by A49,INCPROJ:5;
              hence |{u,v,w}| = 0 by A37,A51,BKMODEL1:77;
              thus 1 < (u`1)^2 + (u`2)^2
              proof
                assume not 1 < (u`1)^2 + (u`2)^2;
                then Dir |[u`1,u`2,1]| in BK_model \/ absolute by Th28;
                hence contradiction by A37,A1,EUCLID_5:3,A34;
              end;
            end;
            hence contradiction by A53,Th24;
          end;
          hence thesis by COLLSP:10;
        end;
      end;
    end;
