let G be IncProjStr ; :: thesis: for a, b, c, d being POINT of G
for P, Q, R being LINE of G st a,b,c,d is_a_quadrangle & {a,b} on P & {a,c} on Q & {a,d} on R holds
P,Q,R are_mutually_distinct
let a, b, c, d be POINT of G; :: thesis: for P, Q, R being LINE of G st a,b,c,d is_a_quadrangle & {a,b} on P & {a,c} on Q & {a,d} on R holds
P,Q,R are_mutually_distinct
let P, Q, R be LINE of G; :: thesis: ( a,b,c,d is_a_quadrangle & {a,b} on P & {a,c} on Q & {a,d} on R implies P,Q,R are_mutually_distinct )
assume that
A1: a,b,c,d is_a_quadrangle and
A2: {a,b} on P and
A3: {a,c} on Q and
A4: {a,d} on R ; :: thesis: P,Q,R are_mutually_distinct
c,d,a is_a_triangle by A1;
then a,c,d is_a_triangle by Th11;
then A5: Q <> R by A3, A4, Th19;
d,a,b is_a_triangle by A1;
then a,d,b is_a_triangle by Th11;
then A6: R <> P by A2, A4, Th19;
a,b,c is_a_triangle by A1;
then P <> Q by A2, A3, Th19;
hence P,Q,R are_mutually_distinct by A5, A6, ZFMISC_1:def 5; :: thesis: verum