let IPP be 2-dimensional Desarguesian IncProjSp; for a, b, c, q being POINT of IPP
for A, B, C, O being LINE of IPP st not a on A & not b on B & not a on C & not b on C & c on A & c on C & a <> b & a on O & b on O & q on O & not q on A & q <> b & not A,B,C are_concurrent & not B,C,O are_concurrent holds
ex Q being LINE of IPP st
( c on Q & not b on Q & not q on Q & (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) )
let a, b, c, q be POINT of IPP; for A, B, C, O being LINE of IPP st not a on A & not b on B & not a on C & not b on C & c on A & c on C & a <> b & a on O & b on O & q on O & not q on A & q <> b & not A,B,C are_concurrent & not B,C,O are_concurrent holds
ex Q being LINE of IPP st
( c on Q & not b on Q & 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 be LINE of IPP; ( not a on A & not b on B & not a on C & not b on C & c on A & c on C & a <> b & a on O & b on O & q on O & not q on A & q <> b & not A,B,C are_concurrent & not B,C,O are_concurrent implies ex Q being LINE of IPP st
( c on Q & not b on Q & 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
and
A2:
not b on B
and
A3:
( not a on C & not b on C )
and
A4:
c on A
and
A5:
c on C
and
A6:
a <> b
and
A7:
( a on O & b on O & q on O )
and
A8:
not q on A
and
A9:
q <> b
and
A10:
( not A,B,C are_concurrent & not B,C,O are_concurrent )
; ex Q being LINE of IPP st
( c on Q & not b on Q & 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
A11:
d on C
and
A12:
d on B
by INCPROJ:def 9;
consider O1 being LINE of IPP such that
A13:
( a on O1 & d on O1 )
by INCPROJ:def 5;
consider O3 being LINE of IPP such that
A14:
( b on O3 & d on O3 )
by INCPROJ:def 5;
consider p being POINT of IPP such that
A15:
p on A
and
A16:
p on O1
by INCPROJ:def 9;
consider O2 being LINE of IPP such that
A17:
( q on O2 & p on O2 )
by INCPROJ:def 5;
consider pp9 being POINT of IPP such that
A18:
pp9 on O3
and
A19:
pp9 on O2
by INCPROJ:def 9;
consider Q being LINE of IPP such that
A20:
c on Q
and
A21:
pp9 on Q
by INCPROJ:def 5;
now ( q <> a implies ex Q being LINE of IPP st
( c on Q & not b on Q & not q on Q & (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) ) )A22:
(
{a,b,q} on O &
{c,pp9} on Q )
by A7, A20, A21, INCSP_1:1, INCSP_1:2;
A23:
{b,d,pp9} on O3
by A14, A18, INCSP_1:2;
A24:
(
{a,d,p} on O1 &
{q,p,pp9} on O2 )
by A13, A16, A17, A19, INCSP_1:2;
assume A25:
q <> a
;
ex Q being LINE of IPP st
( c on Q & not b on Q & not q on Q & (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) )A26:
(
{c,p} on A &
{c,d} on C )
by A4, A5, A11, A15, INCSP_1:1;
then A27:
( not
q on Q & not
b on Q )
by A1, A3, A6, A8, A9, A10, A12, A25, A22, A24, A23, Th18;
Q <> A
by A1, A3, A6, A8, A9, A10, A12, A25, A26, A22, A24, A23, Th18;
then
(IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q))
by A1, A2, A3, A4, A5, A7, A8, A10, A11, A12, A13, A14, A15, A16, A17, A18, A19, A20, A21, A25, A27, Th14;
hence
ex
Q being
LINE of
IPP st
(
c on Q & not
b on Q & not
q on Q &
(IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) )
by A20, A27;
verum end;
hence
ex Q being LINE of IPP st
( c on Q & not b on Q & not q on Q & (IncProj (C,b,B)) * (IncProj (A,a,C)) = (IncProj (Q,b,B)) * (IncProj (A,q,Q)) )
by A3, A5; verum