let FdSp be FanodesSp; :: thesis: for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
( not a,b,c are_collinear & not b,a,d are_collinear & not c,d,a are_collinear & not d,c,b are_collinear & not a,c,b are_collinear & not b,a,c are_collinear & not b,c,a are_collinear & not c,a,b are_collinear & not c,b,a are_collinear & not b,d,a are_collinear & not a,b,d are_collinear & not a,d,b are_collinear & not d,a,b are_collinear & not d,b,a are_collinear & not c,a,d are_collinear & not a,c,d are_collinear & not a,d,c are_collinear & not d,a,c are_collinear & not d,c,a are_collinear & not d,b,c are_collinear & not b,c,d are_collinear & not b,d,c are_collinear & not c,b,d are_collinear & not c,d,b are_collinear )
let a, b, c, d be Element of FdSp; :: thesis: ( parallelogram a,b,c,d implies ( not a,b,c are_collinear & not b,a,d are_collinear & not c,d,a are_collinear & not d,c,b are_collinear & not a,c,b are_collinear & not b,a,c are_collinear & not b,c,a are_collinear & not c,a,b are_collinear & not c,b,a are_collinear & not b,d,a are_collinear & not a,b,d are_collinear & not a,d,b are_collinear & not d,a,b are_collinear & not d,b,a are_collinear & not c,a,d are_collinear & not a,c,d are_collinear & not a,d,c are_collinear & not d,a,c are_collinear & not d,c,a are_collinear & not d,b,c are_collinear & not b,c,d are_collinear & not b,d,c are_collinear & not c,b,d are_collinear & not c,d,b are_collinear ) )
assume A1: parallelogram a,b,c,d ; :: thesis: ( not a,b,c are_collinear & not b,a,d are_collinear & not c,d,a are_collinear & not d,c,b are_collinear & not a,c,b are_collinear & not b,a,c are_collinear & not b,c,a are_collinear & not c,a,b are_collinear & not c,b,a are_collinear & not b,d,a are_collinear & not a,b,d are_collinear & not a,d,b are_collinear & not d,a,b are_collinear & not d,b,a are_collinear & not c,a,d are_collinear & not a,c,d are_collinear & not a,d,c are_collinear & not d,a,c are_collinear & not d,c,a are_collinear & not d,b,c are_collinear & not b,c,d are_collinear & not b,d,c are_collinear & not c,b,d are_collinear & not c,d,b are_collinear )
then A2: ( not c,d,a are_collinear & not d,c,b are_collinear ) by Th27;
( not a,b,c are_collinear & not b,a,d are_collinear ) by A1, Th27;
hence ( not a,b,c are_collinear & not b,a,d are_collinear & not c,d,a are_collinear & not d,c,b are_collinear & not a,c,b are_collinear & not b,a,c are_collinear & not b,c,a are_collinear & not c,a,b are_collinear & not c,b,a are_collinear & not b,d,a are_collinear & not a,b,d are_collinear & not a,d,b are_collinear & not d,a,b are_collinear & not d,b,a are_collinear & not c,a,d are_collinear & not a,c,d are_collinear & not a,d,c are_collinear & not d,a,c are_collinear & not d,c,a are_collinear & not d,b,c are_collinear & not b,c,d are_collinear & not b,d,c are_collinear & not c,b,d are_collinear & not c,d,b are_collinear ) by A2, Th10; :: thesis: verum