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 Th19:
  (for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS st o<>p2 &
  o<>p3 & p2<>p3 & p1<>p2 & p1<>p3 & o<>q2 & o<>q3 & q2<>q3 & q1<>q2 & q1<>q3 &
not o,p1,q1 are_collinear & o,p1,p2 are_collinear & o,p1,p3 are_collinear &
o,q1,
  q2 are_collinear & o,q1,q3 are_collinear & p1,q2,r3 are_collinear & q1,p2,r3
  are_collinear & p1,q3,r2 are_collinear & p3,q1,r2 are_collinear & p2,q3,r1
are_collinear & p3,q2,r1 are_collinear holds r1,r2,r3 are_collinear) implies
for o
,a1,a2,a3,b1,b2,b3,c1,c2,c3 being POINT of IncProjSp_of(CPS) for A1,A2,A3,B1,B2
  ,B3,C1,C2,C3 being LINE of IncProjSp_of(CPS) st o,a1,a2,a3
are_mutually_distinct & o,b1,b2,b3 are_mutually_distinct & A3<>B3 & o on A3 &
  o on B3 & {a2,b3,c1} on A1 & {a3,b1,c2} on B1 & {a1,b2,c3} on C1 & {a1,b3,c2}
on A2 & {a3,b2,c1} on B2 & {a2,b1,c3} on C2 & {b1,b2,b3} on A3 & {a1,a2,a3} on
  B3 & {c1,c2} on C3 holds c3 on C3
proof
  assume
A1: for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS st o<>p2 & o<>p3
& p2<>p3 & p1<>p2 & p1<>p3 & o<>q2 & o<>q3 & q2<>q3 & q1<>q2 & q1<>q3 & not o,
  p1,q1 are_collinear & o,p1,p2 are_collinear & o,p1,p3 are_collinear & o,q1,q2
  are_collinear & o,q1,q3 are_collinear & p1,q2,r3 are_collinear & q1,p2,r3
  are_collinear & p1,q3,r2 are_collinear & p3,q1,r2 are_collinear & p2,q3,r1
  are_collinear & p3,q2,r1 are_collinear holds r1,r2,r3 are_collinear;
  let o,a1,a2,a3,b1,b2,b3,c1,c2,c3 be POINT of IncProjSp_of(CPS);
  let A1,A2,A3,B1,B2,B3,C1,C2,C3 be LINE of IncProjSp_of(CPS) such that
A2: o,a1,a2,a3 are_mutually_distinct and
A3: o,b1,b2,b3 are_mutually_distinct and
A4: A3<>B3 and
A5: o on A3 and
A6: o on B3 and
A7: {a2,b3,c1} on A1 and
A8: {a3,b1,c2} on B1 and
A9: {a1,b2,c3} on C1 and
A10: {a1,b3,c2} on A2 and
A11: {a3,b2,c1} on B2 and
A12: {a2,b1,c3} on C2 and
A13: {b1,b2,b3} on A3 and
A14: {a1,a2,a3} on B3 and
A15: {c1,c2} on C3;
  reconsider o9= o, a19= a1, a29= a2, a39= a3, b19= b1, b29= b2, b39= b3, c19=
  c1, c29= c2, c39= c3 as Point of CPS;
A16: b1 on A3 by A13,INCSP_1:2;
A17: c3 on C1 by A9,INCSP_1:2;
  a1 on C1 & b2 on C1 by A9,INCSP_1:2;
  then
A18: a19,b29,c39 are_collinear by A17,Th10;
A19: c1 on A1 by A7,INCSP_1:2;
A20: o9<>b39 & b29<>b39 by A3,ZFMISC_1:def 6;
A21: a29<>a39 & a19<>a29 by A2,ZFMISC_1:def 6;
A22: b3 on A2 & c2 on A2 by A10,INCSP_1:2;
A23: a1 on A2 by A10,INCSP_1:2;
  then
A24: a19,b39,c29 are_collinear by A22,Th10;
A25: b19<>b29 & b19<>b39 by A3,ZFMISC_1:def 6;
A26: b3 on A1 by A7,INCSP_1:2;
A27: a1 on B3 by A14,INCSP_1:2;
A28: not o9,a19,b19 are_collinear
  proof
A29: o<>a1 by A2,ZFMISC_1:def 6;
    assume not thesis;
    then consider K being LINE of IncProjSp_of(CPS) such that
A30: o on K and
A31: a1 on K and
A32: b1 on K by Th10;
    o<>b1 by A3,ZFMISC_1:def 6;
    then K = A3 by A5,A16,A30,A32,Th8;
    hence contradiction by A4,A6,A27,A30,A31,A29,Th8;
  end;
A33: c3 on C2 by A12,INCSP_1:2;
  a2 on C2 & b1 on C2 by A12,INCSP_1:2;
  then
A34: b19,a29,c39 are_collinear by A33,Th10;
A35: a3 on B1 by A8,INCSP_1:2;
A36: b2 on A3 by A13,INCSP_1:2;
  then
A37: o9,b19,b29 are_collinear by A5,A16,Th10;
A38: a3 on B2 & c1 on B2 by A11,INCSP_1:2;
A39: b2 on B2 by A11,INCSP_1:2;
  then
A40: a39,b29,c19 are_collinear by A38,Th10;
A41: b3 on A3 by A13,INCSP_1:2;
  then
A42: o9,b19,b39 are_collinear by A5,A16,Th10;
A43: a19<>a39 & o9<>b29 by A2,A3,ZFMISC_1:def 6;
A44: o9<>a29 & o9<>a39 by A2,ZFMISC_1:def 6;
A45: c2 on B1 by A8,INCSP_1:2;
A46: a2 on A1 by A7,INCSP_1:2;
  then
A47: a29,b39,c19 are_collinear by A26,A19,Th10;
A48: b1<>b2 by A3,ZFMISC_1:def 6;
A49: a1<>a2 by A2,ZFMISC_1:def 6;
A50: o<>b3 by A3,ZFMISC_1:def 6;
A51: b1 on B1 by A8,INCSP_1:2;
  then
A52: a39,b19,c29 are_collinear by A35,A45,Th10;
A53: a3 on B3 by A14,INCSP_1:2;
  then
A54: o9,a19,a39 are_collinear by A6,A27,Th10;
A55: a2 on B3 by A14,INCSP_1:2;
  then o9,a19,a29 are_collinear by A6,A27,Th10;
  then c19,c29,c39 are_collinear by A1,A44,A21,A43,A20,A25,A28,A54,A37,A42,A18
,A34,A24,A52,A47,A40;
  then
A56: ex K being LINE of IncProjSp_of(CPS) st c1 on K & c2 on K & c3 on K by
Th10;
A57: o<>a3 by A2,ZFMISC_1:def 6;
A58: c1<>c2
  proof
    assume
A59: not thesis;
    not a3 on A3 by A4,A5,A6,A57,A53,Th8;
    then B1<>B2 by A48,A35,A51,A39,A16,A36,Th8;
    then
A60: c1 = a3 by A35,A45,A38,A59,Th8;
    not b3 on B3 by A4,A5,A6,A50,A41,Th8;
    then A1<>A2 by A49,A46,A26,A23,A27,A55,Th8;
    then c1 = b3 by A26,A19,A22,A59,Th8;
    hence contradiction by A4,A5,A6,A50,A41,A53,A60,Th8;
  end;
  c1 on C3 & c2 on C3 by A15,INCSP_1:1;
  hence thesis by A58,A56,Th8;
end;
