let CLSP be CollSp; :: thesis: for a, b, c being Point of CLSP st a,b,c is_collinear holds
( b,a,c is_collinear & a,c,b is_collinear )
let a, b, c be Point of CLSP; :: thesis: ( a,b,c is_collinear implies ( b,a,c is_collinear & a,c,b is_collinear ) )
assume A1: a,b,c is_collinear ; :: thesis: ( b,a,c is_collinear & a,c,b is_collinear )
then ( a = b or ( a <> b & a,b,b is_collinear & a,b,a is_collinear & a,b,c is_collinear ) ) by Th7;
hence b,a,c is_collinear by Th7, Th8; :: thesis: a,c,b is_collinear
( a = b or ( a <> b & a,b,a is_collinear & a,b,c is_collinear & a,b,b is_collinear ) ) by A1, Th7;
hence a,c,b is_collinear by Th7, Th8; :: thesis: verum