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 Th16:
  not a on A & not a on C & not b on C & not b on Q & not A,B,C
are_concurrent & not B,C,O are_concurrent & A<>Q & Q<>C & a<>b & {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<>b & not q on A & not q 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 b on Q and
A5: not A,B,C are_concurrent and
A6: not B,C,O are_concurrent and
A7: A<>Q and
A8: Q<>C and
A9: a<>b 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: d on O1 by A15,INCSP_1:2;
A23: pp9 on Q by A14,INCSP_1:1;
A24: pp9 on O3 by A17,INCSP_1:2;
A25: pp9 on O2 by A16,INCSP_1:2;
A26: a on O1 by A15,INCSP_1:2;
A27: b on O3 by A17,INCSP_1:2;
A28: d on O3 by A17,INCSP_1:2;
A29: q on O by A13,INCSP_1:2;
A30: b on O by A13,INCSP_1:2;
A31: q<>pp9
  proof
    assume not thesis;
    then O3=O by A4,A30,A29,A27,A24,A23,INCPROJ:def 4;
    hence contradiction by A6,A11,A18,A28;
  end;
A32: pp9 on O2 & q on O2 by A16,INCSP_1:2;
A33: pp9 on Q & p on O2 by A14,A16,INCSP_1:1,2;
A34: p on A by A10,INCSP_1:1;
A35: c on Q by A14,INCSP_1:1;
  then
A36: pp9<>d by A8,A19,A18,A23,A21,INCPROJ:def 4;
A37: O1<>O2
  proof
    assume not thesis;
    then
A38: a on O3 by A26,A22,A25,A28,A24,A36,INCPROJ:def 4;
    a on O & b on O by A13,INCSP_1:2;
    then d on O by A9,A28,A27,A38,INCPROJ:def 4;
    hence contradiction by A6,A11,A18;
  end;
A39: p on O1 by A15,INCSP_1:2;
  then
A40: p<>c by A2,A19,A18,A26,A22,A21,INCPROJ:def 4;
A41: not q on Q
  proof
    assume not thesis;
    then p on Q by A33,A32,A31,INCPROJ:def 4;
    hence contradiction by A7,A20,A35,A34,A40,INCPROJ:def 4;
  end;
A42: a on O by A13,INCSP_1:2;
A43: q<>p
  proof
    assume not thesis;
    then O=O1 by A1,A42,A26,A34,A39,A29,INCPROJ:def 4;
    hence contradiction by A6,A11,A18,A22;
  end;
A44: p on O2 by A16,INCSP_1:2;
  pp9<>c by A3,A19,A18,A28,A27,A24,A21,INCPROJ:def 4;
  then
A45: A<>O2 by A7,A20,A35,A25,A23,INCPROJ:def 4;
A46: q on O2 by A16,INCSP_1:2;
A47: p<>d by A5,A11,A18,A34;
  q<>b
  proof
    assume not thesis;
    then O2=O3 by A4,A46,A25,A27,A24,A23,INCPROJ:def 4;
    hence contradiction by A22,A39,A44,A28,A47,A37,INCPROJ:def 4;
  end;
  hence thesis by A26,A34,A39,A46,A44,A45,A37,A43,A41,INCPROJ:def 4;
end;
