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 Th19:
  not a on A & not a on C & not b on B & not b on C & not q on A &
not A,B,C are_concurrent & a<>b & b<>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 b on Q & not q on Q & A<>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 q on A and
A6: not A,B,C are_concurrent and
A7: a<>b and
A8: b<>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 A by A9,INCSP_1:1;
A19: d on B & d on O by A10,A12,INCSP_1:2;
A20: o on B by A10,INCSP_1:2;
A21: o on B by A10,INCSP_1:2;
A22: d on C by A11,INCSP_1:2;
  then
A23: o<>d by A6,A18,A21;
A24: o on O3 by A16,INCSP_1:2;
A25: b<>oo9
  proof
    assume not thesis;
    then
A26: b on O3 by A16,INCSP_1:2;
    q on O3 & b on O by A12,A16,INCSP_1:2;
    then o on O by A8,A17,A24,A26,INCPROJ:def 4;
    then O=B by A19,A20,A23,INCPROJ:def 4;
    hence contradiction by A3,A12,INCSP_1:2;
  end;
A27: oo9 on O2 by A15,INCSP_1:2;
A28: C<>O2 by A4,A15,INCSP_1:2;
A29: o9 on O1 by A14,INCSP_1:2;
A30: oo9 on O2 by A15,INCSP_1:2;
A31: oo9 on Q by A13,INCSP_1:1;
A32: c on Q & b on O2 by A13,A15,INCSP_1:1,2;
A33: c on A by A9,INCSP_1:1;
A34: a on O1 by A14,INCSP_1:2;
A35: b on O by A12,INCSP_1:2;
A36: o on O1 by A14,INCSP_1:2;
  c on C by A11,INCSP_1:2;
  then
A37: o<>c by A6,A33,A21;
  then
A38: o9<>c by A1,A33,A18,A34,A36,A29,INCPROJ:def 4;
A39: not b on Q
  proof
    assume not thesis;
    then
A40: c on O2 by A32,A31,A30,A25,INCPROJ:def 4;
A41: o9 on O2 by A15,INCSP_1:2;
    o9 on C & c on C by A11,INCSP_1:2;
    hence contradiction by A38,A28,A41,A40,INCPROJ:def 4;
  end;
A42: c on Q by A13,INCSP_1:1;
A43: o9 on C by A11,INCSP_1:2;
A44: a on O by A12,INCSP_1:2;
A45: b on O2 by A15,INCSP_1:2;
A46: O1<>O2
  proof
    assume not thesis;
    then o on O by A7,A44,A35,A34,A36,A45,INCPROJ:def 4;
    then O=B by A19,A20,A23,INCPROJ:def 4;
    hence contradiction by A3,A12,INCSP_1:2;
  end;
A47: o on O3 & q on O3 by A16,INCSP_1:2;
A48: oo9 on Q & oo9 on O3 by A13,A16,INCSP_1:1,2;
A49: o9 on O2 by A15,INCSP_1:2;
A50: oo9 on O3 & q on O3 by A16,INCSP_1:2;
A51: d on O by A12,INCSP_1:2;
A52: d on B by A10,INCSP_1:2;
  o9<>d
  proof
    assume not thesis;
    then O1=O by A2,A22,A44,A51,A34,A29,INCPROJ:def 4;
    hence contradiction by A3,A21,A52,A35,A51,A36,A23,INCPROJ:def 4;
  end;
  then O<>O2 by A2,A22,A43,A44,A51,A49,INCPROJ:def 4;
  then
A53: q<>oo9 by A8,A17,A35,A45,A27,INCPROJ:def 4;
A54: not q on Q
  proof
    assume not thesis;
    then Q=O3 by A31,A50,A53,INCPROJ:def 4;
    hence contradiction by A5,A33,A18,A42,A47,A37,INCPROJ:def 4;
  end;
  o<>o9 by A6,A18,A21,A43;
  then o<>oo9 by A36,A29,A49,A27,A46,INCPROJ:def 4;
  hence thesis by A18,A48,A47,A39,A54,INCPROJ:def 4;
end;
