let IPP be Fanoian IncProjSp; ex a, b, c, d being POINT of IPP ex A being LINE of IPP st
( a on A & b on A & c on A & d on A & a,b,c,d are_mutually_distinct )
consider p, q, r, s, a, b, c being POINT of IPP, A, B, C, Q, L, R, S, D being LINE of IPP, d being POINT of IPP such that
not q on L
and
not r on L
and
not p on Q
and
not s on Q
and
not p on R
and
not r on R
and
not q on S
and
not s on S
and
{a,p,s} on L
and
{a,q,r} on Q
and
{b,q,s} on R
and
{b,p,r} on S
and
A1:
{c,p,q} on A
and
{c,r,s} on B
and
{a,b} on C
and
not c on C
and
b on D
and
c on D
and
C,D,R,S are_mutually_distinct
and
A2:
( d on A & c,d,p,q are_mutually_distinct )
by Lm2;
A3:
q on A
by A1, INCSP_1:2;
( c on A & p on A )
by A1, INCSP_1:2;
hence
ex a, b, c, d being POINT of IPP ex A being LINE of IPP st
( a on A & b on A & c on A & d on A & a,b,c,d are_mutually_distinct )
by A2, A3; verum