reserve IPS for IncProjSp,
  z for POINT of IPS;
reserve IPP for Desarguesian 2-dimensional IncProjSp,
  a,b,c,d,p,pp9,q,o,o9,o99 ,oo9 for POINT of IPP,
  r,s,x,y,o1,o2 for POINT of IPP,
  O1,O2,O3,O4,A,B,C,O,Q,Q1 ,Q2,Q3,R,S,X for LINE of IPP;

theorem Th18:
  not a on A & not a on C & not b on C & not q on A & not A,B,C
are_concurrent & not B,C,O are_concurrent & a<>b & b<>q & q<>a & {c,p} on A & d
on B & {c,d} on C & {a,b,q} on O & {c,pp9} on Q & {a,d,p} on O1 & {q,p,pp9} on
  O2 & {b,d,pp9} on O3 implies Q<>A & Q<>C & not q on Q & not b on Q
proof
  assume that
A1: not a on A and
A2: not a on C and
A3: not b on C and
A4: not q on A and
A5: not A,B,C are_concurrent and
A6: not B,C,O are_concurrent and
A7: a<>b and
A8: b<>q and
A9: q<>a and
A10: {c,p} on A and
A11: d on B and
A12: {c,d} on C and
A13: {a,b,q} on O and
A14: {c,pp9} on Q and
A15: {a,d,p} on O1 and
A16: {q,p,pp9} on O2 and
A17: {b,d,pp9} on O3;
A18: d on C by A12,INCSP_1:1;
A19: c on C by A12,INCSP_1:1;
A20: c on A by A10,INCSP_1:1;
  then
A21: c <>d by A5,A11,A19;
A22: pp9 on O3 by A17,INCSP_1:2;
A23: p on O2 by A16,INCSP_1:2;
A24: pp9 on Q by A14,INCSP_1:1;
A25: q on O by A13,INCSP_1:2;
A26: a on O by A13,INCSP_1:2;
A27: b on O3 by A17,INCSP_1:2;
A28: d on O1 by A15,INCSP_1:2;
  then
A29: O<>O1 by A6,A11,A18;
A30: p on O1 by A15,INCSP_1:2;
A31: p on O2 by A16,INCSP_1:2;
A32: a on O1 by A15,INCSP_1:2;
A33: pp9 on O2 by A16,INCSP_1:2;
A34: q on O2 by A16,INCSP_1:2;
A35: p on A by A10,INCSP_1:1;
  then
A36: p<>d by A5,A11,A18;
A37: pp9<>d
  proof
    assume not thesis;
    then
A38: q on O1 by A28,A30,A34,A31,A33,A36,INCPROJ:def 4;
    a on O & q on O by A13,INCSP_1:2;
    hence contradiction by A9,A32,A29,A38,INCPROJ:def 4;
  end;
A39: d on O3 by A17,INCSP_1:2;
A40: b on O by A13,INCSP_1:2;
A41: p<>pp9
  proof
    assume not thesis;
    then O1=O3 by A28,A30,A39,A22,A36,INCPROJ:def 4;
    hence contradiction by A7,A26,A40,A32,A27,A29,INCPROJ:def 4;
  end;
A42: c on Q by A14,INCSP_1:1;
  then
A43: O3<>Q by A3,A19,A18,A39,A27,A21,INCPROJ:def 4;
A44: not b on Q
  proof
    assume not thesis;
    then
A45: b on O2 by A33,A27,A22,A24,A43,INCPROJ:def 4;
A46: q on O by A13,INCSP_1:2;
    q on O2 & b on O by A13,A16,INCSP_1:2;
    then p on O by A8,A23,A45,A46,INCPROJ:def 4;
    hence contradiction by A1,A26,A32,A35,A30,A29,INCPROJ:def 4;
  end;
  p<>c by A2,A19,A18,A32,A28,A30,A21,INCPROJ:def 4;
  then
A47: O2<>Q by A4,A20,A42,A35,A34,A31,INCPROJ:def 4;
A48: O<>O3 by A6,A11,A18,A39;
  not q on Q
  proof
    assume not thesis;
    then q=pp9 by A34,A33,A24,A47,INCPROJ:def 4;
    hence contradiction by A8,A40,A25,A27,A22,A48,INCPROJ:def 4;
  end;
  hence thesis by A18,A35,A34,A31,A33,A39,A27,A22,A24,A41,A37,A44,INCPROJ:def 4
;
end;
