let G be IncProjStr ; :: thesis: ( G is configuration iff for p, q being POINT of G
for P, Q being LINE of G st {p,q} on P & {p,q} on Q & not p = q holds
P = Q )

hereby :: thesis: ( ( for p, q being POINT of G
for P, Q being LINE of G st {p,q} on P & {p,q} on Q & not p = q holds
P = Q ) implies G is configuration )

assume A1: G is configuration ; :: thesis: for p, q being POINT of G
for P, Q being LINE of G st {p,q} on P & {p,q} on Q & not p = q holds
P = Q
let p, q be POINT of G; :: thesis: for P, Q being LINE of G st {p,q} on P & {p,q} on Q & not p = q holds
P = Q
let P, Q be LINE of G; :: thesis: ( {p,q} on P & {p,q} on Q & not p = q implies P = Q )
assume that
A2: {p,q} on P and
A3: {p,q} on Q ; :: thesis: ( p = q or P = Q )
A4: ( p on Q & q on Q ) by A3, INCSP_1:1;
( p on P & q on P ) by A2, INCSP_1:1;
hence ( p = q or P = Q ) by A1, A4, Def4; :: thesis: verum

end;

hereby :: thesis: verum

assume A5: for p, q being POINT of G
for P, Q being LINE of G st {p,q} on P & {p,q} on Q & not p = q holds
P = Q ; :: thesis: G is configuration

now

let p, q be POINT of G; :: thesis: for P, Q being LINE of G st p on P & q on P & p on Q & q on Q & not p = q holds
P = Q
let P, Q be LINE of G; :: thesis: ( p on P & q on P & p on Q & q on Q & not p = q implies P = Q )
assume ( p on P & q on P & p on Q & q on Q ) ; :: thesis: ( p = q or P = Q )
then ( {p,q} on P & {p,q} on Q ) by INCSP_1:1;
hence ( p = q or P = Q ) by A5; :: thesis: verum

end;

hence G is configuration by Def4; :: thesis: verum

end;