
theorem
  for p,q being Element of absolute
  for a,b being Element of BK_model
  holds ex N being invertible Matrix of 3,F_Real st
   homography(N).:absolute = absolute &
  (homography(N)).a = b & (homography(N)).p = q
  proof
    let p,q be Element of absolute;
    let a,b be Element of BK_model;
    consider p9 be Element of absolute such that
A1: p <> p9 and
A2: p,a,p9 are_collinear by Th16;
    consider q9 be Element of absolute such that
A3: q <> q9 and
A4: q,b,q9 are_collinear by Th16;
    consider t be Element of real_projective_plane such that
A5: tangent p /\ tangent p9 = {t} by A1,Th25;
A6: t in tangent p /\ tangent p9 by A5,TARSKI:def 1;
    consider u be Element of real_projective_plane such that
A7: tangent q /\ tangent q9 = {u} by A3,Th25;
A8: u in tangent q /\ tangent q9 by A7,TARSKI:def 1;
    reconsider a9 = a as Element of real_projective_plane;
    p <> p9 & a in BK_model & a,p,p9 are_collinear &
      t in tangent p & t in tangent p9 by A1,A2,A6,XBOOLE_0:def 4,COLLSP:4;
    then consider Ra be Element of real_projective_plane such that
A9: Ra in absolute and
A10: a9,t,Ra are_collinear by Th31;
    reconsider RRa = Ra as Element of absolute by A9;
    reconsider b9 = b as Element of real_projective_plane;
    q <> q9 & b in BK_model & b,q,q9 are_collinear &
      u in tangent q & u in tangent q9 by A3,A4,A8,XBOOLE_0:def 4,COLLSP:4;
    then consider Rb be Element of real_projective_plane such that
A11: Rb in absolute and
A12: b9,u,Rb are_collinear by Th31;
    reconsider RRb = Rb as Element of absolute by A11;
A13: p,p9,Ra are_mutually_distinct
    proof
      now
        consider ra be Element of real_projective_plane such that
A14:    ra = Ra & tangent RRa = Line(ra,pole_infty RRa) by Def04;
        thus p <> Ra
        proof
          assume p = Ra;
          then t in Line(ra,pole_infty RRa) by A14,A6,XBOOLE_0:def 4;
          then ra,pole_infty RRa,t are_collinear by COLLSP:11;
          then
A15:      ra,t,pole_infty RRa are_collinear by COLLSP:4;
A16:      ra,t,a9 are_collinear by A14,A10,HESSENBE:1;
          ra <> t
          proof
            assume ra = t;
            then t in absolute & t in tangent p & t in tangent p9
              by A14,A6,XBOOLE_0:def 4;
            then t in tangent p /\ absolute & t in tangent p9 /\ absolute
              by XBOOLE_0:def 4;
            then t in {p} & t in {p9} by Th22;
            then t = p & t = p9 by TARSKI:def 1;
            hence contradiction by A1;
          end;
          then a in tangent RRa & a in BK_model by A16,A15,A14,COLLSP:6,11;
          then tangent RRa meets BK_model by XBOOLE_0:def 4;
          hence contradiction by Th30;
        end;
        thus p9 <> Ra
        proof
          assume p9 = Ra;
          then t in Line(ra,pole_infty RRa) by A14,A6,XBOOLE_0:def 4;
          then ra,pole_infty RRa,t are_collinear by COLLSP:11;
          then
A17:      ra,t,pole_infty RRa are_collinear by COLLSP:4;
A18:      ra,t,a9 are_collinear by A14,A10,HESSENBE:1;
          ra <> t
          proof
            assume ra = t;
            then t in absolute & t in tangent p & t in tangent p9
              by A14,A6,XBOOLE_0:def 4;
            then t in tangent p /\ absolute & t in tangent p9 /\ absolute
              by XBOOLE_0:def 4;
            then t in {p} & t in {p9} by Th22;
            then t = p & t = p9 by TARSKI:def 1;
            hence contradiction by A1;
          end;
          then a in tangent RRa & a in BK_model by A18,A17,A14,COLLSP:6,11;
          then tangent RRa meets BK_model by XBOOLE_0:def 4;
          hence contradiction by Th30;
        end;
      end;
      hence thesis by A1;
    end;
    now
      now
        consider rb be Element of real_projective_plane such that
A19:    rb = Rb & tangent RRb = Line(rb,pole_infty RRb) by Def04;
        thus q <> Rb
        proof
          assume q = Rb;
          then u in Line(rb,pole_infty RRb) by A19,A8,XBOOLE_0:def 4;
          then rb,pole_infty RRb,u are_collinear by COLLSP:11;
          then
A20:      rb,u,pole_infty RRb are_collinear by COLLSP:4;
A21:      rb,u,b9 are_collinear by A19,A12,HESSENBE:1;
          rb <> u
          proof
            assume rb = u;
            then u in absolute & u in tangent q & u in tangent q9
              by A19,A8,XBOOLE_0:def 4;
            then u in tangent q /\ absolute & u in tangent q9 /\ absolute
              by XBOOLE_0:def 4;
            then u in {q} & u in {q9} by Th22;
            then u = q & u = q9 by TARSKI:def 1;
            hence contradiction by A3;
          end;
          then b in tangent RRb & b in BK_model by A21,A20,A19,COLLSP:6,11;
          then tangent RRb meets BK_model by XBOOLE_0:def 4;
          hence contradiction by Th30;
        end;
        thus q9 <> Rb
        proof
          assume q9 = Rb;
          then u in Line(rb,pole_infty RRb) by A19,A8,XBOOLE_0:def 4;
          then rb,pole_infty RRb,u are_collinear by COLLSP:11;
          then
A22:      rb,u,pole_infty RRb are_collinear by COLLSP:4;
A23:      rb,u,b9 are_collinear by A19,A12,HESSENBE:1;
          rb <> u
          proof
            assume rb = u;
            then u in absolute & u in tangent q & u in tangent q9
              by A19,A8,XBOOLE_0:def 4;
            then u in tangent q /\ absolute & u in tangent q9 /\ absolute
              by XBOOLE_0:def 4;
            then u in {q} & u in {q9} by Th22;
            then u = q & u = q9 by TARSKI:def 1;
            hence contradiction by A3;
          end;
          then b in tangent RRb & b in BK_model by A23,A22,A19,COLLSP:6,11;
          then tangent RRb meets BK_model by XBOOLE_0:def 4;
          hence contradiction by Th30;
        end;
      end;
      hence q,q9,Rb are_mutually_distinct by A3;
    end;
    then
    consider N be invertible Matrix of 3,F_Real such that
A24: homography(N).:absolute = absolute and
A25: (homography(N)).p = q and
A26: (homography(N)).p9 = q9 and
A27: (homography(N)).Ra = Rb and
A28: (homography(N)).t = u by A9,A11,A6,A8,A13,Th38;
    reconsider plp = p, plq = p9, plr = Ra, pls = t,
      plt = a, np = q, nq = q9, nr = Rb, ns = u, nu = b
      as Element of real_projective_plane;
    now
      thus plp <> plq by A1;
      thus np <> nq by A3;
      thus nr <> ns
      proof
        consider rb be Element of real_projective_plane such that
A29:    rb = Rb & tangent RRb = Line(rb,pole_infty RRb) by Def04;
        rb <> u
        proof
          assume rb = u;
          then u in absolute & u in tangent q & u in tangent q9
            by A29,A8,XBOOLE_0:def 4;
          then u in tangent q /\ absolute & u in tangent q9 /\ absolute
            by XBOOLE_0:def 4;
          then u in {q} & u in {q9} by Th22;
          then u = q & u = q9 by TARSKI:def 1;
          hence contradiction by A3;
        end;
        hence thesis by A29;
      end;
      thus plr <> pls
      proof
        consider ra be Element of real_projective_plane such that
A30:    ra = Ra & tangent RRa = Line(ra,pole_infty RRa) by Def04;
        ra <> t
        proof
          assume ra = t;
          then t in absolute & t in tangent p & t in tangent p9
            by A30,A6,XBOOLE_0:def 4;
          then t in tangent p /\ absolute & t in tangent p9 /\ absolute
            by XBOOLE_0:def 4;
          then t in {p} & t in {p9} by Th22;
          then t = p & t = p9 by TARSKI:def 1;
          hence contradiction by A1;
        end;
        hence thesis by A30;
      end;
      thus plp,plq,plt are_collinear by A2,COLLSP:4;
      thus plr,pls,plt are_collinear by A10,HESSENBE:1;
      thus np = homography(N).plp & nq = homography(N).plq &
        nr = homography(N).plr & ns = homography(N).pls by A25,A26,A27,A28;
      thus np,nq,nu are_collinear by A4,HESSENBE:1;
      thus nr,ns,nu are_collinear by A12,HESSENBE:1;
      thus not plp,plq,plr are_collinear
      proof
        assume plp,plq,plr are_collinear;
        then p,p9,RRa are_collinear;
        hence contradiction by A13,BKMODEL1:92;
      end;
    end;
    then nu = homography(N).plt by Th43;
    hence thesis by A24,A25;
  end;
