
theorem Th44:
  for P,Q being Element of BK_model st P <> Q holds
  ex P1,P2,P3,P4 being Element of absolute,
              P5 being Element of ProjectiveSpace TOP-REAL 3 st
  P1 <> P2 &
  P,Q,P1 are_collinear & P,Q,P2 are_collinear &
  P,P5,P3 are_collinear & Q,P5,P4 are_collinear &
  P1,P2,P3 are_mutually_distinct &
  P1,P2,P4 are_mutually_distinct &
  P5 in tangent P1 /\ tangent P2
  proof
    let P,Q being Element of BK_model;
    assume
A1: P <> Q;
    then consider P1,P2 be Element of absolute such that
A2: P1 <> P2 and
A3: P,Q,P1 are_collinear and
A4: P,Q,P2 are_collinear by Th12;
    consider R be Element of real_projective_plane such that
A5: R in tangent P1 & R in tangent P2 by Th24;
    consider u be Element of TOP-REAL 3 such that
A6: u is non zero and
A7: R = Dir u by ANPROJ_1:26;
    per cases;
    suppose u.3 = 0;
      reconsider RR = R as Element of ProjectiveSpace TOP-REAL 3;
      P,P1,P2 are_collinear by A1,A3,A4,COLLSP:6;
      then consider PT1 be Element of ProjectiveSpace TOP-REAL 3 such that
A8:   PT1 in absolute and
A9:   P,RR,PT1 are_collinear by A2,A5,Th31;
      Q,P,P1 are_collinear & Q,P,P2 are_collinear by A3,A4,COLLSP:4;
      then Q,P1,P2 are_collinear by A1,COLLSP:6;
      then consider PT2 be Element of ProjectiveSpace TOP-REAL 3 such that
A10:  PT2 in absolute and
A11:  Q,RR,PT2 are_collinear by A2,A5,Th31;
      now
        thus P,Q,P1 are_collinear by A3;
        thus P,Q,P2 are_collinear by A4;
A12:    PT1 <> RR
        proof
          assume PT1 = RR;
          then PT1 in absolute /\ tangent P1 & PT1 in absolute /\ tangent P2
            by A5,A8,XBOOLE_0:def 4;
          then PT1 in {P1} & PT1 in {P2} by Th22;
          then PT1 = P1 & PT1 = P2 by TARSKI:def 1;
          hence contradiction by A2;
        end;
A13:    PT2 <> RR
        proof
          assume PT2 = RR;
          then PT2 in absolute /\ tangent P1 & PT2 in absolute /\ tangent P2
            by A5,A10,XBOOLE_0:def 4;
          then PT2 in {P1} & PT2 in {P2} by Th22;
          then PT2 = P1 & PT2 = P2 by TARSKI:def 1;
          hence contradiction by A2;
        end;
A14:    P2 <> PT1
        proof
          P,PT1,RR are_collinear by A9,COLLSP:4;
          hence thesis by A5,A12,Th32;
        end;
        P1 <> PT1
        proof
          assume
A15:      P1 = PT1;
          consider p1 be Element of real_projective_plane such that
A16:      p1 = P1 & tangent P1 = Line(p1,pole_infty P1) by Def04;
          reconsider pt1 = PT1,
                     rr = RR,
                     p = P as Element of real_projective_plane;
A17:      p1,pole_infty P1,pt1 are_collinear &
            p1,pole_infty P1,rr are_collinear
            by A15,A5,Th21,A16,COLLSP:11;
          rr,pt1,p are_collinear by A9,COLLSP:8;
          then P in tangent P1 & P in BK_model by A12,A17,A16,COLLSP:9,11;
          then tangent P1 meets BK_model by XBOOLE_0:def 4;
          hence contradiction by Th30;
        end;
        hence P1,P2,PT1 are_mutually_distinct by A14,A2;
A18:    P1 <> PT2
        proof
          Q,PT2,RR are_collinear by A11,COLLSP:4;
          hence thesis by A5,A13,Th32;
        end;
        P2 <> PT2
        proof
          assume
A19:      P2 = PT2;
          consider p2 be Element of real_projective_plane such that
A20:      p2 = P2 & tangent P2 = Line(p2,pole_infty P2) by Def04;
          reconsider pt2 = PT2,
                     rr = RR,
                     q = Q as Element of real_projective_plane;
A21:      p2,pole_infty P2,pt2 are_collinear &
            p2,pole_infty P2,rr are_collinear
            by A20,A19,A5,Th21,COLLSP:11;
          rr,pt2,q are_collinear by A11,COLLSP:8;
          then Q in tangent P2 & Q in BK_model by A20,A13,A21,COLLSP:9,11;
          then tangent P2 meets BK_model by XBOOLE_0:def 4;
          hence contradiction by Th30;
        end;
        hence P1,P2,PT2 are_mutually_distinct by A18,A2;
        thus R in tangent P1 /\ tangent P2 by A5,XBOOLE_0:def 4;
        thus P,RR,PT1 are_collinear by A9;
        thus Q,RR,PT2 are_collinear by A11;
      end;
      hence thesis by A8,A10;
    end;
    suppose
A22:  u.3 <> 0;
      reconsider v = |[u.1 / u.3,u.2 / u.3,1]| as
        non zero Element of TOP-REAL 3 by BKMODEL1:41;
A24:  u.3 * (u.1 / u.3) = u.1 & u.3 * (u.2/u.3) = u.2 by A22,XCMPLX_1:87;
      u.3 * v = |[ u.3 * (u.1/u.3), u.3 * (u.2/u.3),u.3 * 1]| by EUCLID_5:8
             .= |[ u`1,u.2,u.3 ]| by A24,EUCLID_5:def 1
             .= |[ u`1,u`2,u.3 ]| by EUCLID_5:def 2
             .= |[ u`1,u`2,u`3 ]| by EUCLID_5:def 3
             .= u by EUCLID_5:3;
      then are_Prop v,u by A22,ANPROJ_1:1;
      then
A25:  R = Dir v & v.3 = 1 by A6,A7,ANPROJ_1:22;
      reconsider RR = R as Element of ProjectiveSpace TOP-REAL 3;
      P <> RR
      proof
        assume P = RR;
        then BK_model meets tangent P1 by A5,XBOOLE_0:def 4;
        hence contradiction by Th30;
      end;
      then consider PT1 be Element of absolute such that
A26:  P,RR,PT1 are_collinear by A25,Th03;
      Q <> RR
      proof
        assume Q = RR;
        then BK_model meets tangent P2 by A5,XBOOLE_0:def 4;
        hence contradiction by Th30;
      end;
      then consider PT2 being Element of absolute such that
A27:  Q,RR,PT2 are_collinear by A25,Th03;
      now
        thus P,Q,P1 are_collinear by A3;
        thus P,Q,P2 are_collinear by A4;
A28:    PT1 <> RR
        proof
          assume PT1 = RR;
          then PT1 in absolute /\ tangent P1 & PT1 in absolute /\ tangent P2
            by A5,XBOOLE_0:def 4;
          then PT1 in {P1} & PT1 in {P2} by Th22;
          then PT1 = P1 & PT1 = P2 by TARSKI:def 1;
          hence contradiction by A2;
        end;
A29:    PT2 <> RR
        proof
          assume PT2 = RR;
          then PT2 in absolute /\ tangent P1 & PT2 in absolute /\ tangent P2
            by A5,XBOOLE_0:def 4;
          then PT2 in {P1} & PT2 in {P2} by Th22;
          then PT2 = P1 & PT2 = P2 by TARSKI:def 1;
          hence contradiction by A2;
        end;
A30:    P2 <> PT1
        proof
          P,PT1,RR are_collinear by A26,COLLSP:4;
          hence thesis by A5,A28,Th32;
        end;
        P1 <> PT1
        proof
          assume
A31:      P1 = PT1;
          consider p1 be Element of real_projective_plane such that
A32:      p1 = P1 & tangent P1 = Line(p1,pole_infty P1) by Def04;
          reconsider pt1 = PT1,
                     rr = RR,
                     p = P as Element of real_projective_plane;
A33:      p1,pole_infty P1,pt1 are_collinear &
            p1,pole_infty P1,rr are_collinear by A31,A5,Th21,A32,COLLSP:11;
          rr,pt1,p are_collinear by A26,COLLSP:8;
          then P in tangent P1 & P in BK_model by A28,A33,A32,COLLSP:9,11;
          then tangent P1 meets BK_model by XBOOLE_0:def 4;
          hence contradiction by Th30;
        end;
        hence P1,P2,PT1 are_mutually_distinct by A30,A2;
A34:    P1 <> PT2
        proof
          Q,PT2,RR are_collinear by A27,COLLSP:4;
          hence thesis by A5,A29,Th32;
        end;
        P2 <> PT2
        proof
          assume
A35:      P2 = PT2;
          consider p2 being Element of real_projective_plane such that
A36:      p2 = P2 & tangent P2 = Line(p2,pole_infty P2) by Def04;
          reconsider pt2 = PT2,
                     rr = RR,
                     q = Q as Element of real_projective_plane;
A37:      p2,pole_infty P2,pt2 are_collinear &
            p2,pole_infty P2,rr are_collinear by A35,A5,Th21,A36,COLLSP:11;
          rr,pt2,q are_collinear by A27,COLLSP:8;
          then Q in tangent P2 by A36,A29,A37,COLLSP:9,11;
          then tangent P2 meets BK_model by XBOOLE_0:def 4;
          hence contradiction by Th30;
        end;
        hence P1,P2,PT2 are_mutually_distinct by A34,A2;
        thus RR in tangent P1 /\ tangent P2 by A5,XBOOLE_0:def 4;
        thus P,RR,PT1 are_collinear by A26;
        thus Q,RR,PT2 are_collinear by A27;
      end;
      hence thesis;
    end;
  end;
