let FdSp be FanodesSp; for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
not a,d '||' b,c
let a, b, c, d be Element of FdSp; ( parallelogram a,b,c,d implies not a,d '||' b,c )
assume A1:
parallelogram a,b,c,d
; not a,d '||' b,c
then
not a,b,c are_collinear
;
then A2:
not a,b '||' a,c
;
( a,b '||' c,d & a,c '||' b,d )
by A1;
then
not b,c '||' a,d
by A2, Def1;
hence
not a,d '||' b,c
by PARSP_1:19; verum