reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;
reserve x,y,z,t,u,w for Element of AS;
reserve K,X,Y,Z,X9,Y9 for Subset of AS;
reserve a,b,c,d,p,q,r,p9 for POINT of IncProjSp_of(AS);
reserve A for LINE of IncProjSp_of(AS);
reserve A,K,M,N,P,Q for LINE of IncProjSp_of(AS);

theorem Th44:
  for a,b,c,d,p being Element of the Points of ProjHorizon(AS),M,N
,P,Q being Element of the Lines of ProjHorizon(AS) st a on M & b on M & c on N
  & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P &
not p on Q & M<>N ex q being Element of the Points of ProjHorizon(AS) st q on P
  & q on Q
proof
  let a,b,c,d,p be Element of the Points of ProjHorizon(AS),M,N,P,Q be Element
  of the Lines of ProjHorizon(AS) such that
A1: a on M and
A2: b on M and
A3: c on N and
A4: d on N and
A5: p on M and
A6: p on N and
A7: a on P and
A8: c on P and
A9: b on Q and
A10: d on Q and
A11: not p on P and
A12: not p on Q and
A13: M<>N;
  reconsider M9=[M,2],N9=[N,2],P9=[P,2],Q9=[Q,2] as LINE of IncProjSp_of(AS)
  by Th25;
  reconsider a9=a,b9=b,c9=c,d9=d,p9=p as POINT of IncProjSp_of(AS) by Th22;
A14: b9 on M9 by A2,Th37;
A15: M9<>N9
  proof
    assume M9=N9;
    then M = [N,2]`1
      .= N;
    hence contradiction by A13;
  end;
A16: d9 on N9 by A4,Th37;
A17: c9 on N9 by A3,Th37;
A18: c9 on P9 by A8,Th37;
A19: a9 on P9 by A7,Th37;
A20: p9 on N9 by A6,Th37;
A21: p9 on M9 by A5,Th37;
A22: not p9 on Q9 by A12,Th37;
A23: not p9 on P9 by A11,Th37;
A24: d9 on Q9 by A10,Th37;
A25: b9 on Q9 by A9,Th37;
  a9 on M9 by A1,Th37;
  then consider q9 being POINT of IncProjSp_of(AS) such that
A26: q9 on P9 and
A27: q9 on Q9 by A14,A17,A16,A21,A20,A19,A18,A25,A24,A23,A22,A15,Lm12;
  [q9,[P,2]] in the Inc of IncProjSp_of(AS) by A26,INCSP_1:def 1;
  then reconsider q=q9 as Element of the Points of ProjHorizon(AS) by Th42;
  take q;
  thus thesis by A26,A27,Th37;
end;
