reserve CPS for proper CollSp,
  B for Subset of CPS,
  p for Point of CPS,
  x, y, z, Y for set;
reserve a,b,c,p,q for POINT of IncProjSp_of(CPS),
  P,Q for LINE of IncProjSp_of(CPS),
  a9,b9,c9,p9,q9,r9 for Point of CPS,
  P9 for LINE of CPS;
reserve CPS for CollProjectiveSpace,
  a,b,c,d,p,q for POINT of IncProjSp_of(CPS ),
  P,Q,S,M,N for LINE of IncProjSp_of(CPS),
  a9,b9,c9,d9,p9,q9 for Point of CPS;

theorem Th17:
  (for p1, r2, q, r1, q1, p, r being Point of CPS holds (p1,r2,q
  are_collinear & r1,q1,q are_collinear & p1,r1,p are_collinear & r2,q1,p
are_collinear & p1,q1,r are_collinear & r2,r1,r are_collinear &
p,q,r are_collinear
  implies (p1,r2,q1 are_collinear or p1,r2,r1 are_collinear or p1,r1,q1
are_collinear or r2,r1,q1 are_collinear))) implies
for p, q, r, s, a, b, c being
POINT of IncProjSp_of(CPS) for L, Q, R, S, A, B, C being LINE of IncProjSp_of(
CPS) st not q on L & not r on L & not p on Q & not s on Q & not p on R & not r
on R & {a,p,s} on L & {a,q,r} on Q & {b,q,s} on R & {b,p,r} on S & {c,p,q} on A
  & {c,r,s} on B & {a,b} on C holds not c on C
proof
  assume
A1: for p1, r2, q, r1, q1, p, r being Element of CPS holds (p1,r2,q
  are_collinear & r1,q1,q are_collinear & p1,r1,p are_collinear & r2,q1,p
are_collinear & p1,q1,r are_collinear & r2,r1,r are_collinear &
p,q,r are_collinear
  implies (p1,r2,q1 are_collinear or p1,r2,r1 are_collinear or p1,r1,q1
  are_collinear or r2,r1,q1 are_collinear));
  let p, q, r, s, a, b, c be POINT of IncProjSp_of(CPS);
  let L, Q, R, S, A, B, C be LINE of IncProjSp_of(CPS) such that
A2: not q on L and
A3: not r on L and
A4: not p on Q and
A5: not s on Q and
A6: not p on R and
A7: not r on R and
A8: {a,p,s} on L and
A9: {a,q,r} on Q and
A10: {b,q,s} on R and
A11: {b,p,r} on S and
A12: {c,p,q} on A and
A13: {c,r,s} on B and
A14: {a,b} on C;
  reconsider p9=p, q9=q, r9=r, s9=s, a9=a, b9=b, c9=c as Point of CPS;
A15: p on L & s on L by A8,INCSP_1:2;
A16: s on R by A10,INCSP_1:2;
A17: now
    assume p9,r9,s9 are_collinear;
    then
A18: ex K being LINE of IncProjSp_of(CPS) st p on K & r on K & s on K by Th10;
    hence s on S by A3,A6,A15,A16,Th8;
    thus contradiction by A3,A6,A15,A16,A18,Th8;
  end;
A19: now
    assume p9,s9,q9 are_collinear;
    then
A20: ex K being LINE of IncProjSp_of(CPS) st p on K & s on K & q on K by Th10;
    hence p on R by A2,A15,A16,Th8;
    thus contradiction by A2,A6,A15,A16,A20,Th8;
  end;
  a on L by A8,INCSP_1:2;
  then
A21: p9,s9,a9 are_collinear by A15,Th10;
  assume
A22: not thesis;
  a on C & b on C by A14,INCSP_1:1;
  then
A23: a9,b9,c9 are_collinear by A22,Th10;
A24: q on Q & r on Q by A9,INCSP_1:2;
A25: q on R by A10,INCSP_1:2;
A26: now
    assume p9,r9,q9 are_collinear;
    then
A27: ex K being LINE of IncProjSp_of(CPS) st p on K & r on K & q on K by Th10;
    hence q on S by A4,A7,A24,A25,Th8;
    thus contradiction by A4,A7,A24,A25,A27,Th8;
  end;
A28: now
    assume r9,s9,q9 are_collinear;
    then
A29: ex K being LINE of IncProjSp_of(CPS) st r on K & s on K & q on K by Th10;
    hence r on R by A5,A24,A25,Th8;
    thus contradiction by A5,A7,A24,A25,A29,Th8;
  end;
  a on Q by A9,INCSP_1:2;
  then
A30: r9,q9,a9 are_collinear by A24,Th10;
A31: r on S by A11,INCSP_1:2;
  b on S & p on S by A11,INCSP_1:2;
  then
A32: p9,r9,b9 are_collinear by A31,Th10;
A33: s on B by A13,INCSP_1:2;
  c on B & r on B by A13,INCSP_1:2;
  then
A34: r9,s9,c9 are_collinear by A33,Th10;
A35: q on A by A12,INCSP_1:2;
  c on A & p on A by A12,INCSP_1:2;
  then
A36: p9,q9,c9 are_collinear by A35,Th10;
  b on R by A10,INCSP_1:2;
  then s9,q9,b9 are_collinear by A25,A16,Th10;
  hence contradiction by A1,A23,A32,A21,A30,A36,A34,A26,A17,A19,A28;
end;
