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 Th17:
  not a on A & not a on C & not b on B & not b on C & not b on Q &
not A,B,C are_concurrent & a<>b & A<>Q & {c,o} on A & {o,o99,d} on B & {c,d,o9}
on C & {a,b,d} on O & {c,oo9} on Q & {a,o,o9} on O1 & {b,o9,oo9} on O2 & {o,oo9
  ,q} on O3 & q on O implies not q on A & not q on Q & b<>q
proof
  assume that
A1: not a on A and
A2: not a on C and
A3: not b on B and
A4: not b on C and
A5: not b on Q and
A6: not A,B,C are_concurrent and
A7: a<>b and
A8: A<>Q and
A9: {c,o} on A and
A10: {o,o99,d} on B and
A11: {c,d,o9} on C and
A12: {a,b,d} on O and
A13: {c,oo9} on Q and
A14: {a,o,o9} on O1 and
A15: {b,o9,oo9} on O2 and
A16: {o,oo9,q} on O3 and
A17: q on O;
A18: o on B by A10,INCSP_1:2;
A19: c on A & c on Q by A9,A13,INCSP_1:1;
A20: o on A by A9,INCSP_1:1;
A21: oo9 on O2 by A15,INCSP_1:2;
A22: b on O2 & oo9 on O2 by A15,INCSP_1:2;
A23: oo9 on Q by A13,INCSP_1:1;
A24: b on O2 by A15,INCSP_1:2;
A25: o on O1 by A14,INCSP_1:2;
A26: o on A by A9,INCSP_1:1;
A27: d on C by A11,INCSP_1:2;
  then
A28: o<>d by A6,A26,A18;
A29: d on B by A10,INCSP_1:2;
A30: b on O by A12,INCSP_1:2;
A31: oo9 on Q by A13,INCSP_1:1;
A32: o9 on O2 by A15,INCSP_1:2;
A33: q on O3 by A16,INCSP_1:2;
A34: a on O1 by A14,INCSP_1:2;
A35: d on O by A12,INCSP_1:2;
A36: a on O by A12,INCSP_1:2;
A37: O1<>O2
  proof
    assume not thesis;
    then o on O by A7,A36,A30,A34,A25,A24,INCPROJ:def 4;
    hence contradiction by A3,A18,A29,A30,A35,A28,INCPROJ:def 4;
  end;
A38: o on O3 & oo9 on O3 by A16,INCSP_1:2;
A39: o9 on C by A11,INCSP_1:2;
  then
A40: o<>o9 by A6,A26,A18;
A41: b<>q
  proof
    assume not thesis;
    then
A42: o on O2 by A5,A23,A22,A38,A33,INCPROJ:def 4;
A43: o9 on O2 by A15,INCSP_1:2;
    o on O1 & o9 on O1 by A14,INCSP_1:2;
    hence contradiction by A37,A40,A42,A43,INCPROJ:def 4;
  end;
A44: c on A by A9,INCSP_1:1;
A45: c on C by A11,INCSP_1:2;
  then
A46: o<>c by A6,A44,A18;
A47: o9 on O1 by A14,INCSP_1:2;
  then o9<>c by A1,A44,A26,A34,A25,A46,INCPROJ:def 4;
  then
A48: c <>oo9 by A4,A45,A39,A24,A32,A21,INCPROJ:def 4;
  o9<>d
  proof
    assume not thesis;
    then O1=O by A2,A27,A36,A35,A34,A47,INCPROJ:def 4;
    hence contradiction by A3,A18,A29,A30,A35,A25,A28,INCPROJ:def 4;
  end;
  then O<>O2 by A2,A27,A39,A36,A35,A32,INCPROJ:def 4;
  then
A49: q<>oo9 by A5,A17,A30,A31,A24,A21,INCPROJ:def 4;
A50: c on Q by A13,INCSP_1:1;
A51: not q on Q
  proof
    assume not thesis;
    then o on Q by A23,A38,A33,A49,INCPROJ:def 4;
    hence contradiction by A8,A44,A26,A50,A46,INCPROJ:def 4;
  end;
A52: q<>o by A3,A17,A18,A29,A30,A35,A28,INCPROJ:def 4;
  not q on A
  proof
    assume not thesis;
    then oo9 on A by A20,A38,A33,A52,INCPROJ:def 4;
    hence contradiction by A8,A19,A23,A48,INCPROJ:def 4;
  end;
  hence thesis by A51,A41;
end;
