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
  AS is not AffinPlane implies ProjHorizon(AS) is IncProjSp
proof
  set XX = ProjHorizon(AS);
A1: for a,b being Element of the Points of XX ex A being Element of the
  Lines of XX st a on A & b on A by Th40;
A2: for A being Element of the Lines of XX ex a,b,c being Element of the
  Points of XX st a<>b & b<>c & c <>a & a on A & b on A & c on A
  proof
    let A be Element of the Lines of XX;
    consider a,b,c being Element of the Points of XX such that
A3: a on A and
A4: b on A and
A5: c on A and
A6: a<>b and
A7: b<>c and
A8: c <>a by Th39;
    take a,b,c;
    thus thesis by A3,A4,A5,A6,A7,A8;
  end;
  assume AS is not AffinPlane;
  then
A9: ex a being Element of the Points of XX, A being Element of the Lines of
  XX st not a on A by Lm1;
A10: for a,b,c,d,p,q being Element of the Points of XX, M,N,P,Q being
Element of the Lines of XX 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 holds ex
  q being Element of the Points of XX st q on P & q on Q by Th44;
  for a,b being Element of the Points of XX, A,K being Element of the
  Lines of XX st a on A & b on A & a on K & b on K holds a=b or A=K by Th38;
  hence thesis by A1,A9,A2,A10,INCPROJ:def 4,def 5,def 6,def 7,def 8;
end;
