let SAS be Semi_Affine_Space; :: thesis: for a, b, c, a9, b9, c9 being Element of SAS st a,b,c is_collinear & b <> c & parallelogram a,a9,b,b9 & parallelogram a,a9,c,c9 holds
parallelogram b,b9,c,c9
let a, b, c, a9, b9, c9 be Element of SAS; :: thesis: ( a,b,c is_collinear & b <> c & parallelogram a,a9,b,b9 & parallelogram a,a9,c,c9 implies parallelogram b,b9,c,c9 )
assume that
A1: a,b,c is_collinear and
A2: b <> c and
A3: parallelogram a,a9,b,b9 and
A4: parallelogram a,a9,c,c9 ; :: thesis: parallelogram b,b9,c,c9
A5: b <> b9 by A3, Th54;
a,b // a,c by A1, Def2;
then A6: a,b // b,c by Th18;
( not a,a9,b is_collinear & a,a9 // b,b9 ) by A3, Def3;
hence parallelogram b,b9,c,c9 by A2, A3, A4, A6, A5, Th39, Th68; :: thesis: verum