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 Th40:
  for a,b being Element of the Points of ProjHorizon(AS) ex A
  being Element of the Lines of ProjHorizon(AS) st a on A & b on A
proof
  let a,b be Element of the Points of ProjHorizon(AS);
  consider X such that
A1: a=LDir(X) and
A2: X is being_line by Th14;
  consider X9 such that
A3: b=LDir(X9) and
A4: X9 is being_line by Th14;
  consider x,y being Element of AS such that
A5: x in X9 and
  y in X9 and
  x<>y by A4,AFF_1:19;
A6: x in x*X by A2,AFF_4:def 3;
  x*X is being_line by A2,AFF_4:27;
  then consider Z such that
A7: X9 c= Z and
A8: x*X c= Z and
A9: Z is being_plane by A4,A5,A6,AFF_4:38;
A10: X9 '||' Z by A4,A7,A9,AFF_4:42;
  reconsider A=PDir(Z) as Element of the Lines of ProjHorizon(AS) by A9,Th15;
  take A;
  X // x*X by A2,AFF_4:def 3;
  then X '||' Z by A2,A8,A9,AFF_4:41;
  hence thesis by A1,A2,A3,A4,A9,A10,Th36;
end;
