
theorem Th09:
  for L being LINE of IncProjSp_of real_projective_plane
  for P being Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3
  st P = Dir u & P in L & u.3 <> 0 holds
  ex Q being Element of ProjectiveSpace TOP-REAL 3 st
  (ex v being non zero Element of TOP-REAL 3 st
  Q = Dir v & Q in L & P <> Q & v.3 <> 0)
  proof
    let L be LINE of IncProjSp_of real_projective_plane;
    let P be Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume that
A1: P = Dir u and
A2: P in L and
A3: u.3 <> 0;
    assume
A4: not ex Q being Element of ProjectiveSpace TOP-REAL 3 st
      (ex v being non zero Element of TOP-REAL 3 st
      Q = Dir v & Q in L & P <> Q & v.3 <> 0);
    consider p,q be Element of ProjectiveSpace TOP-REAL 3 such that
A5: p <> q and
A6: L = Line(p,q) by BKMODEL1:73;
    consider up be Element of TOP-REAL 3 such that
A7: up is not zero and
A8: p = Dir up by ANPROJ_1:26;
    consider vp be Element of TOP-REAL 3 such that
A9: vp is not zero and
A10: q = Dir vp by ANPROJ_1:26;
    reconsider P9 = P as POINT of IncProjSp_of real_projective_plane
      by INCPROJ:3;
    reconsider L9 = L as LINE of real_projective_plane by INCPROJ:4;
    per cases;
    suppose
A11:  up`3 = 0 & vp`3 = 0;
      per cases by A5;
      suppose
A12:    P <> p;
A13:    u`3 <> 0 by A3,EUCLID_5:def 3;
        |[u`1+up`1,u`2+up`2,u`3]| is non zero
        proof
          assume |[u`1+up`1,u`2+up`2,u`3]| is zero;
          then u`3 = 0 by EUCLID_5:4,FINSEQ_1:78;
          hence contradiction by A3,EUCLID_5:def 3;
        end;
        then
        reconsider wp = |[u`1+up`1,u`2+up`2,u`3]| as
          non zero Element of TOP-REAL 3;
        reconsider R = Dir wp as Element of ProjectiveSpace TOP-REAL 3
          by ANPROJ_1:26;
A14:    |{u,up,wp}| = 0 by A11,Th08;
        now
          thus R <> P
          proof
            assume R = P;
            then are_Prop wp,u by A1,ANPROJ_1:22;
            then consider a be Real such that a <> 0 and
A15:        wp = a * u by ANPROJ_1:1;
            a = 1 & up`1 = 0 & up`2 = 0
            proof
A16:          |[a * u`1,a * u`2, a * u`3 ]|
                = |[u`1+up`1,u`2 + up`2,u`3]| by A15,EUCLID_5:7;
              then a * u`1 = u`1 + up`1 & a * u`2 = u`2 + up`2 &
                a * u`3 = u`3 by FINSEQ_1:78;
              hence a = 1 by XCMPLX_1:7,A13;
              then u`1 = u`1 + up`1 & u`2 = u`2 + up`2 by A16,FINSEQ_1:78;
              hence thesis;
            end;
            hence contradiction by A7,A11,EUCLID_5:3,4;
          end;
          reconsider R2 = R as POINT of IncProjSp_of real_projective_plane
            by INCPROJ:2;
          now
            L = Line(P,p)
            proof
              P in L & p in L & P <> p & L is LINE of real_projective_plane
                by A6,A2,A12,COLLSP:10,INCPROJ:4;
              hence thesis by COLLSP:19;
            end;
            hence R2 on L by A1,A7,A8,A14,BKMODEL1:77;
            thus L is LINE of real_projective_plane by INCPROJ:4;
          end;
          hence R in L by INCPROJ:5;
          thus wp.3 <> 0 by A13;
        end;
        hence contradiction by A4;
      end;
      suppose
A17:    P <> q;
A18:    u`3 <> 0 by A3,EUCLID_5:def 3;
        |[u`1+vp`1,u`2+vp`2,u`3]| is non zero
        proof
          assume |[u`1+vp`1,u`2+vp`2,u`3]| is zero;
          then u`3 = 0 by EUCLID_5:4,FINSEQ_1:78;
          hence contradiction by A3,EUCLID_5:def 3;
        end;
        then reconsider wp = |[u`1 + vp`1,u`2 + vp`2,u`3]| as
          non zero Element of TOP-REAL 3;
        reconsider R = Dir wp as Element of ProjectiveSpace TOP-REAL 3
          by ANPROJ_1:26;
A19:    |{u,vp,wp}| = 0 by A11,Th08;
        now
          thus R <> P
          proof
            assume R = P;
            then are_Prop wp,u by A1,ANPROJ_1:22;
            then consider a be Real such that a <> 0 and
A20:        wp = a * u by ANPROJ_1:1;
            a = 1 & vp`1 = 0 & vp`2 = 0
            proof
              |[a * u`1,a * u`2, a * u`3 ]|
                = |[u`1+vp`1,u`2 + vp`2,u`3]| by A20,EUCLID_5:7;
              then
A21:          a * u`1 = u`1 + vp`1 & a * u`2 = u`2 + vp`2 &
                a * u`3 = u`3 by FINSEQ_1:78;
              hence a = 1 by XCMPLX_1:7,A18;
              hence thesis by A21;
            end;
            hence contradiction by A11,A9,EUCLID_5:3,4;
          end;
          reconsider R2 = R as POINT of IncProjSp_of real_projective_plane
            by INCPROJ:2;
          now
            L = Line(P,q)
            proof
              P in L & q in L & P <> q &
                L is LINE of real_projective_plane
                  by A6,A2,COLLSP:10,A17,INCPROJ:4;
              hence thesis by COLLSP:19;
            end;
            hence R2 on L by A1,A9,A10,A19,BKMODEL1:77;
            thus L is LINE of real_projective_plane by INCPROJ:4;
          end;
          hence R in L by INCPROJ:5;
          thus wp.3 <> 0 by A18;
        end;
        hence contradiction by A4;
      end;
    end;
    suppose up`3 <> 0 or vp`3 <> 0;
      then per cases;
      suppose
A22:    up`3 <> 0;
        per cases;
        suppose
A23:      P = p;
          per cases;
          suppose
A24:        vp`3 <> 0;
            vp.3 = 0 by A9,A10,A23,A5,A4,A6,COLLSP:10;
            hence contradiction by A24,EUCLID_5:def 3;
          end;
          suppose
A25:        vp`3 = 0;
A26:        u`3 <> 0 by A3,EUCLID_5:def 3;
            |[u`1+vp`1,u`2+vp`2,u`3]| is non zero
            proof
              assume |[u`1+vp`1,u`2+vp`2,u`3]| is zero;
              then u`3 = 0 by EUCLID_5:4,FINSEQ_1:78;
              hence contradiction by A3,EUCLID_5:def 3;
            end;
            then reconsider wp = |[u`1 + vp`1,u`2 + vp`2,u`3]| as
              non zero Element of TOP-REAL 3;
            reconsider R = Dir wp as Element of ProjectiveSpace TOP-REAL 3
              by ANPROJ_1:26;
A27:        |{u,vp,wp}| = 0 by A25,Th08;
            now
              thus R <> P
              proof
                assume R = P;
                then are_Prop wp,u by A1,ANPROJ_1:22;
                then consider a be Real such that a <> 0 and
A28:            wp = a * u by ANPROJ_1:1;
                a = 1 & vp`1 = 0 & vp`2 = 0
                proof
                  |[a * u`1,a * u`2, a * u`3 ]|
                    = |[u`1+vp`1,u`2 + vp`2,u`3]| by A28,EUCLID_5:7;
                  then
A29:              a * u`1 = u`1 + vp`1 & a * u`2 = u`2 + vp`2 &
                    a * u`3 = u`3 by FINSEQ_1:78;
                  hence a = 1 by XCMPLX_1:7,A26;
                  hence thesis by A29;
                end;
                hence contradiction by A25,EUCLID_5:3,4,A9;
              end;
              reconsider R2 = R as POINT of IncProjSp_of real_projective_plane
                by INCPROJ:2;
              R2 on L & L is LINE of real_projective_plane
                by A6,A23,A1,A9,A10,A27,BKMODEL1:77,INCPROJ:4;
              hence R in L by INCPROJ:5;
              thus wp.3 <> 0 by A26;
            end;
            hence contradiction by A4;
          end;
        end;
        suppose P <> p;
          then up.3 = 0 by A8,A6,A4,A7,COLLSP:10;
          hence contradiction by A22,EUCLID_5:def 3;
        end;
      end;
      suppose
A30:    vp`3 <> 0;
        per cases;
        suppose
A31:      P = q;
          per cases;
          suppose
A32:        up`3 <> 0;
            up.3 = 0 by A7,A8,A31,A5,A4,A6,COLLSP:10;
            hence contradiction by A32,EUCLID_5:def 3;
          end;
          suppose
A33:        up`3 = 0;
A34:        u`3 <> 0 by A3,EUCLID_5:def 3;
           |[u`1+up`1,u`2+up`2,u`3]| is non zero
           proof
             assume |[u`1+up`1,u`2+up`2,u`3]| is zero;
             then u`3 = 0 by EUCLID_5:4,FINSEQ_1:78;
             hence contradiction by A3,EUCLID_5:def 3;
           end;
           then reconsider wp = |[u`1 + up`1,u`2 + up`2,u`3]| as
             non zero Element of TOP-REAL 3;
           reconsider R = Dir wp as Element of ProjectiveSpace TOP-REAL 3
             by ANPROJ_1:26;
A35:       |{u,up,wp}| = 0 by A33,Th08;
           now
              thus R <> P
              proof
                assume R = P;
                then are_Prop wp,u by A1,ANPROJ_1:22;
                then consider a be Real such that a <> 0 and
A36:            wp = a * u by ANPROJ_1:1;
                a = 1 & up`1 = 0 & up`2 = 0
                proof
                  |[a * u`1,a * u`2, a * u`3 ]| = |[u`1+up`1,u`2 + up`2,u`3]|
                    by A36,EUCLID_5:7;
                  then
A37:              a * u`1 = u`1 + up`1 & a * u`2 = u`2 + up`2 &
                    a * u`3 = u`3 by FINSEQ_1:78;
                  hence a = 1 by XCMPLX_1:7,A34;
                  hence thesis by A37;
                end;
                hence contradiction by A33,EUCLID_5:3,4,A7;
              end;
              reconsider R2 = R as
                POINT of IncProjSp_of real_projective_plane by INCPROJ:2;
              now
                L = Line(P,p)
                proof
                  P in L & p in L & P <> p &
                    L is LINE of real_projective_plane
                    by A6,A31,A5,COLLSP:10,INCPROJ:4;
                  hence thesis by COLLSP:19;
                end;
                hence R2 on L by A1,A7,A8,A35,BKMODEL1:77;
                thus L is LINE of real_projective_plane by INCPROJ:4;
              end;
              hence R in L by INCPROJ:5;
              thus wp.3 <> 0 by A34;
            end;
            hence contradiction by A4;
          end;
        end;
        suppose P <> q;
          then vp.3 = 0 by A10,A6,COLLSP:10,A4,A9;
          hence contradiction by A30,EUCLID_5:def 3;
        end;
      end;
    end;
  end;
