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 is_collinear & not b,a,d is_collinear & not c,d,a is_collinear & not d,c,b is_collinear & not a,c,b is_collinear & not b,a,c is_collinear & not b,c,a is_collinear & not c,a,b is_collinear & not c,b,a is_collinear & not b,d,a is_collinear & not a,b,d is_collinear & not a,d,b is_collinear & not d,a,b is_collinear & not d,b,a is_collinear & not c,a,d is_collinear & not a,c,d is_collinear & not a,d,c is_collinear & not d,a,c is_collinear & not d,c,a is_collinear & not d,b,c is_collinear & not b,c,d is_collinear & not b,d,c is_collinear & not c,b,d is_collinear & not c,d,b is_collinear )
let a, b, c, d be Element of FdSp; :: thesis: ( parallelogram a,b,c,d implies ( not a,b,c is_collinear & not b,a,d is_collinear & not c,d,a is_collinear & not d,c,b is_collinear & not a,c,b is_collinear & not b,a,c is_collinear & not b,c,a is_collinear & not c,a,b is_collinear & not c,b,a is_collinear & not b,d,a is_collinear & not a,b,d is_collinear & not a,d,b is_collinear & not d,a,b is_collinear & not d,b,a is_collinear & not c,a,d is_collinear & not a,c,d is_collinear & not a,d,c is_collinear & not d,a,c is_collinear & not d,c,a is_collinear & not d,b,c is_collinear & not b,c,d is_collinear & not b,d,c is_collinear & not c,b,d is_collinear & not c,d,b is_collinear ) )
assume A1: parallelogram a,b,c,d ; :: thesis: ( not a,b,c is_collinear & not b,a,d is_collinear & not c,d,a is_collinear & not d,c,b is_collinear & not a,c,b is_collinear & not b,a,c is_collinear & not b,c,a is_collinear & not c,a,b is_collinear & not c,b,a is_collinear & not b,d,a is_collinear & not a,b,d is_collinear & not a,d,b is_collinear & not d,a,b is_collinear & not d,b,a is_collinear & not c,a,d is_collinear & not a,c,d is_collinear & not a,d,c is_collinear & not d,a,c is_collinear & not d,c,a is_collinear & not d,b,c is_collinear & not b,c,d is_collinear & not b,d,c is_collinear & not c,b,d is_collinear & not c,d,b is_collinear )
then A2: not a,b,c is_collinear by Th35;
A3: not b,a,d is_collinear by A1, Th35;
A4: not c,d,a is_collinear by A1, Th35;
not d,c,b is_collinear by A1, Th35;
hence ( not a,b,c is_collinear & not b,a,d is_collinear & not c,d,a is_collinear & not d,c,b is_collinear & not a,c,b is_collinear & not b,a,c is_collinear & not b,c,a is_collinear & not c,a,b is_collinear & not c,b,a is_collinear & not b,d,a is_collinear & not a,b,d is_collinear & not a,d,b is_collinear & not d,a,b is_collinear & not d,b,a is_collinear & not c,a,d is_collinear & not a,c,d is_collinear & not a,d,c is_collinear & not d,a,c is_collinear & not d,c,a is_collinear & not d,b,c is_collinear & not b,c,d is_collinear & not b,d,c is_collinear & not c,b,d is_collinear & not c,d,b is_collinear ) by A2, A3, A4, Th15; :: thesis: verum