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 Th18:
  (for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS st o<>q1 &
p1<>q1 & o<>q2 & p2<>q2 & o<>q3 & p3<>q3 & not o,p1,p2 are_collinear &
not o,p1,
  p3 are_collinear & not o,p2,p3 are_collinear & p1,p2,r3 are_collinear &
q1,q2,r3
  are_collinear & p2,p3,r1 are_collinear & q2,q3,r1 are_collinear & p1,p3,r2
  are_collinear & q1,q3,r2 are_collinear & o,p1,q1 are_collinear & o,p2,q2
are_collinear & o,p3,q3 are_collinear holds r1,r2,r3 are_collinear) implies
for o,
b1,a1,b2,a2,b3,a3,r,s,t being POINT of IncProjSp_of(CPS) for C1,C2,C3,A1,A2,A3,
B1,B2,B3 being LINE of IncProjSp_of(CPS) st {o,b1,a1} on C1 & {o,a2,b2} on C2 &
{o,a3,b3} on C3 & {a3,a2,t} on A1 & {a3,r,a1} on A2 & {a2,s,a1} on A3 & {t,b2,
b3} on B1 & {b1,r,b3} on B2 & {b1,s,b2} on B3 & C1,C2,C3 are_mutually_distinct
  & o<>a1 & o<>a2 & o<>a3 & o<>b1 & o<>b2 & o<>b3 & a1<>b1 & a2<>b2 & a3<>b3
  holds ex O being LINE of IncProjSp_of(CPS) st {r,s,t} on O
proof
  assume
A1: for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Element of CPS st o<>q1 & p1
<>q1 & o<>q2 & p2<>q2 & o<>q3 & p3<>q3 & not o,p1,p2 are_collinear &
not o,p1,p3
  are_collinear & not o,p2,p3 are_collinear & p1,p2,r3 are_collinear & q1,q2,r3
  are_collinear & p2,p3,r1 are_collinear & q2,q3,r1 are_collinear & p1,p3,r2
  are_collinear & q1,q3,r2 are_collinear & o,p1,q1 are_collinear & o,p2,q2
  are_collinear & o,p3,q3 are_collinear holds r1,r2,r3 are_collinear;
  let o,b1,a1,b2,a2,b3,a3,r,s,t be POINT of IncProjSp_of(CPS);
  let C1,C2,C3,A1,A2,A3,B1,B2,B3 be LINE of IncProjSp_of(CPS) such that
A2: {o,b1,a1} on C1 and
A3: {o,a2,b2} on C2 and
A4: {o,a3,b3} on C3 and
A5: {a3,a2,t} on A1 and
A6: {a3,r,a1} on A2 and
A7: {a2,s,a1} on A3 and
A8: {t,b2,b3} on B1 and
A9: {b1,r,b3} on B2 and
A10: {b1,s,b2} on B3 and
A11: C1,C2,C3 are_mutually_distinct and
A12: o<>a1 & o<>a2 & o<>a3 and
A13: o<>b1 and
A14: o<>b2 and
A15: o<>b3 and
A16: a1<>b1 & a2<>b2 & a3<>b3;
  reconsider o9=o,b19=b1,a19=a1,b29=b2,a29=a2,b39=b3,a39=a3,r9=r,s9=s,t9=t as
  Element of CPS;
A17: o on C2 & b2 on C2 by A3,INCSP_1:2;
A18: s on B3 by A10,INCSP_1:2;
  b2 on B3 & b1 on B3 by A10,INCSP_1:2;
  then
A19: b19,b29,s9 are_collinear by A18,Th10;
A20: r on B2 by A9,INCSP_1:2;
  b3 on B2 & b1 on B2 by A9,INCSP_1:2;
  then
A21: b19,b39,r9 are_collinear by A20,Th10;
A22: t on B1 by A8,INCSP_1:2;
  b3 on B1 & b2 on B1 by A8,INCSP_1:2;
  then
A23: b29,b39,t9 are_collinear by A22,Th10;
A24: s on A3 by A7,INCSP_1:2;
  a2 on A3 & a1 on A3 by A7,INCSP_1:2;
  then
A25: a19,a29,s9 are_collinear by A24,Th10;
A26: o on C3 & b3 on C3 by A4,INCSP_1:2;
  a3 on C3 by A4,INCSP_1:2;
  then
A27: o9,b39,a39 are_collinear by A26,Th10;
  a2 on C2 by A3,INCSP_1:2;
  then
A28: o9,b29,a29 are_collinear by A17,Th10;
A29: r on A2 by A6,INCSP_1:2;
  a3 on A2 & a1 on A2 by A6,INCSP_1:2;
  then
A30: a19,a39,r9 are_collinear by A29,Th10;
A31: t on A1 by A5,INCSP_1:2;
  a3 on A1 & a2 on A1 by A5,INCSP_1:2;
  then
A32: a29,a39,t9 are_collinear by A31,Th10;
A33: o on C1 & b1 on C1 by A2,INCSP_1:2;
A34: not o9,b19,b29 are_collinear & not o9,b19,b39 are_collinear & not o9,b29,
  b39 are_collinear
  proof
A35: now
      assume o9,b19,b39 are_collinear;
      then consider K being LINE of IncProjSp_of(CPS) such that
A36:  o on K & b1 on K & b3 on K by Th10;
      K = C1 & K = C3 by A13,A15,A33,A26,A36,Th8;
      hence contradiction by A11,ZFMISC_1:def 5;
    end;
A37: now
      assume o9,b19,b29 are_collinear;
      then consider K being LINE of IncProjSp_of(CPS) such that
A38:  o on K & b1 on K & b2 on K by Th10;
      K = C1 & K = C2 by A13,A14,A33,A17,A38,Th8;
      hence contradiction by A11,ZFMISC_1:def 5;
    end;
A39: now
      assume o9,b29,b39 are_collinear;
      then consider K being LINE of IncProjSp_of(CPS) such that
A40:  o on K & b2 on K & b3 on K by Th10;
      K = C2 & K = C3 by A14,A15,A17,A26,A40,Th8;
      hence contradiction by A11,ZFMISC_1:def 5;
    end;
    assume not thesis;
    hence contradiction by A37,A35,A39;
  end;
  a1 on C1 by A2,INCSP_1:2;
  then o9,b19,a19 are_collinear by A33,Th10;
  then t9,r9,s9 are_collinear
  by A1,A12,A16,A19,A25,A21,A30,A23,A32,A28,A27,A34;
  then consider O being LINE of IncProjSp_of(CPS) such that
A41: t on O & r on O & s on O by Th10;
  {r,s,t} on O by A41,INCSP_1:2;
  hence thesis;
end;
