let S be OAffinSpace; :: thesis: for x, y, z, t being Element of S st x,y,z is_collinear & x,y,t is_collinear holds
x,y '||' z,t
let x, y, z, t be Element of S; :: thesis: ( x,y,z is_collinear & x,y,t is_collinear implies x,y '||' z,t )
assume A1: ( x,y,z is_collinear & x,y,t is_collinear ) ; :: thesis: x,y '||' z,t

now

A2: ( x,y '||' x,z & x,y '||' x,t ) by A1, Def5;
assume x <> y ; :: thesis: x,y '||' z,t
then x,z '||' x,t by A2, Th28;
then z,x '||' z,t by Th26;
then x,z '||' z,t by Th27;
hence x,y '||' z,t by A2, Th28; :: thesis: verum

end;

hence x,y '||' z,t by Th25; :: thesis: verum