let G be IncProjSp; :: thesis:
let a, b, c, d be POINT of G; :: thesis:
assume that
A1: a,b,c,d is_a_quadrangle and
A2: not a,b,c,d are_mutually_distinct ; :: thesis:

now :: thesis:

per cases by A2, ZFMISC_1:def 6;

case a = b ; :: thesis:

then not a,b,c are_mutually_distinct by ZFMISC_1:def 5;
then not a,b,c is_a_triangle by Th26;
hence contradiction by A1; :: thesis:

end;

case b = c ; :: thesis:

then not a,b,c are_mutually_distinct by ZFMISC_1:def 5;
then not a,b,c is_a_triangle by Th26;
hence contradiction by A1; :: thesis:

end;

case c = a ; :: thesis:

then not a,b,c are_mutually_distinct by ZFMISC_1:def 5;
then not a,b,c is_a_triangle by Th26;
hence contradiction by A1; :: thesis:

end;

case d = a ; :: thesis:

then not d,a,b are_mutually_distinct by ZFMISC_1:def 5;
then not d,a,b is_a_triangle by Th26;
hence contradiction by A1; :: thesis:

end;

case d = b ; :: thesis:

then not d,a,b are_mutually_distinct by ZFMISC_1:def 5;
then not d,a,b is_a_triangle by Th26;
hence contradiction by A1; :: thesis:

end;

case d = c ; :: thesis:

then not b,c,d are_mutually_distinct by ZFMISC_1:def 5;
then not b,c,d is_a_triangle by Th26;
hence contradiction by A1; :: thesis:

end;

end;

end;

hence contradiction ; :: thesis: