let IPP be 2-dimensional Desarguesian IncProjSp; :: thesis: for a, b being POINT of IPP
for A, B, C, O, Q being LINE of IPP st not a on A & not b on B & not a on C & not b on C & not A,B,C are_concurrent & A,C,Q are_concurrent & not b on Q & A <> Q & a <> b & a on O & b on O & B,C,O are_concurrent holds
ex q being POINT of IPP st
( q on O & not q on A & not q on Q & (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) )

let a, b be POINT of IPP; :: thesis: for A, B, C, O, Q being LINE of IPP st not a on A & not b on B & not a on C & not b on C & not A,B,C are_concurrent & A,C,Q are_concurrent & not b on Q & A <> Q & a <> b & a on O & b on O & B,C,O are_concurrent holds
ex q being POINT of IPP st
( q on O & not q on A & not q on Q & (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) )

let A, B, C, O, Q be LINE of IPP; :: thesis: ( not a on A & not b on B & not a on C & not b on C & not A,B,C are_concurrent & A,C,Q are_concurrent & not b on Q & A <> Q & a <> b & a on O & b on O & B,C,O are_concurrent implies ex q being POINT of IPP st
( q on O & not q on A & not q on Q & (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) ) )

assume that
A1: ( not a on A & not b on B & not a on C & not b on C & not A,B,C are_concurrent ) and
A2: A,C,Q are_concurrent and
A3: ( not b on Q & A <> Q & a <> b ) and
A4: ( a on O & b on O ) and
A5: B,C,O are_concurrent ; :: thesis: ex q being POINT of IPP st
( q on O & not q on A & not q on Q & (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) )

consider d being POINT of IPP such that
A6: d on B and
A7: d on C and
A8: d on O by A5;
A9: {a,b,d} on O by ;
consider o being POINT of IPP such that
A10: o on A and
A11: o on B by INCPROJ:def 9;
consider O1 being LINE of IPP such that
A12: ( o on O1 & a on O1 ) by INCPROJ:def 5;
consider o9 being POINT of IPP such that
A13: o9 on C and
A14: o9 on O1 by INCPROJ:def 9;
A15: {a,o,o9} on O1 by ;
consider c being POINT of IPP such that
A16: c on A and
A17: c on C and
A18: c on Q by A2;
A19: {c,o} on A by ;
A20: {c,d,o9} on C by ;
A21: {c,o} on A by ;
A22: {c,d,o9} on C by ;
consider O2 being LINE of IPP such that
A23: ( b on O2 & o9 on O2 ) by INCPROJ:def 5;
consider oo9 being POINT of IPP such that
A24: oo9 on Q and
A25: oo9 on O2 by INCPROJ:def 9;
A26: {b,o9,oo9} on O2 by ;
consider o99 being POINT of IPP such that
A27: o99 on B and
o99 on O2 by INCPROJ:def 9;
consider O3 being LINE of IPP such that
A28: ( o on O3 & oo9 on O3 ) by INCPROJ:def 5;
A29: {o,o99,d} on B by ;
A30: {b,o9,oo9} on O2 by ;
A31: {a,o,o9} on O1 by ;
A32: {a,b,d} on O by ;
consider q being POINT of IPP such that
A33: q on O3 and
A34: q on O by INCPROJ:def 9;
A35: {o,oo9,q} on O3 by ;
A36: {c,oo9} on Q by ;
A37: {o,o99,d} on B by ;
then A38: ( not q on A & not q on Q ) by A1, A3, A34, A19, A22, A9, A36, A15, A26, A35, Th17;
A39: {o,oo9,q} on O3 by ;
A40: {c,oo9} on Q by ;
b <> q by A1, A3, A34, A19, A37, A22, A9, A36, A15, A26, A35, Th17;
then (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) by A1, A3, A34, A21, A29, A20, A32, A40, A31, A30, A39, Th15;
hence ex q being POINT of IPP st
( q on O & not q on A & not q on Q & (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) ) by ; :: thesis: verum