let CLSP be CollSp; :: thesis: for a, b, c being Point of CLSP st a,b,c are_collinear holds
( b,a,c are_collinear & a,c,b are_collinear )
let a, b, c be Point of CLSP; :: thesis: ( a,b,c are_collinear implies ( b,a,c are_collinear & a,c,b are_collinear ) )
assume A1: a,b,c are_collinear ; :: thesis: ( b,a,c are_collinear & a,c,b are_collinear )
then ( a = b or ( a <> b & a,b,b are_collinear & a,b,a are_collinear & a,b,c are_collinear ) ) by Th2;
hence b,a,c are_collinear by Th2, Th3; :: thesis: a,c,b are_collinear
( a = b or ( a <> b & a,b,a are_collinear & a,b,c are_collinear & a,b,b are_collinear ) ) by A1, Th2;
hence a,c,b are_collinear by Th2, Th3; :: thesis: verum