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 Th39:
  for A being Element of the Lines of ProjHorizon(AS) ex a,b,c
being Element of the Points of ProjHorizon(AS) st a on A & b on A & c on A & a
  <>b & b<>c & c <>a
proof
  let A be Element of the Lines of ProjHorizon(AS);
  consider X such that
A1: A=PDir(X) and
A2: X is being_plane by Th15;
  consider x,y,z such that
A3: x in X and
A4: y in X and
A5: z in X and
A6: not LIN x,y,z by A2,AFF_4:34;
A7: y<>z by A6,AFF_1:7;
  then
A8: Line(y,z) is being_line by AFF_1:def 3;
  then
A9: Line(y,z) '||' X by A2,A4,A5,A7,AFF_4:19,42;
A10: z<>x by A6,AFF_1:7;
  then
A11: Line(x,z) is being_line by AFF_1:def 3;
  then
A12: Line(x,z) '||' X by A2,A3,A5,A10,AFF_4:19,42;
A13: x<>y by A6,AFF_1:7;
  then
A14: Line(x,y) is being_line by AFF_1:def 3;
  then reconsider
  a=LDir(Line(x,y)),b=LDir(Line(y,z)),c =LDir(Line(x,z)) as Element
  of the Points of ProjHorizon(AS) by A8,A11,Th14;
  take a,b,c;
  Line(x,y) '||' X by A2,A3,A4,A13,A14,AFF_4:19,42;
  hence a on A & b on A & c on A by A1,A2,A14,A8,A11,A9,A12,Th36;
A15: x in Line(x,y) by AFF_1:15;
A16: z in Line(y,z) by AFF_1:15;
A17: y in Line(x,y) by AFF_1:15;
A18: y in Line(y,z) by AFF_1:15;
  thus a<>b
  proof
    assume not thesis;
    then Line(x,y) // Line(y,z) by A14,A8,Th11;
    then z in Line(x,y) by A17,A18,A16,AFF_1:45;
    hence contradiction by A6,A14,A15,A17,AFF_1:21;
  end;
A19: z in Line(x,z) by AFF_1:15;
A20: x in Line(x,z) by AFF_1:15;
  thus b<>c
  proof
    assume not thesis;
    then Line(y,z) // Line(x,z) by A8,A11,Th11;
    then x in Line(y,z) by A16,A20,A19,AFF_1:45;
    hence contradiction by A6,A8,A18,A16,AFF_1:21;
  end;
  thus c <>a
  proof
    assume not thesis;
    then Line(x,y) // Line(x,z) by A14,A11,Th11;
    then z in Line(x,y) by A15,A20,A19,AFF_1:45;
    hence contradiction by A6,A14,A15,A17,AFF_1:21;
  end;
end;
