let p1, p2, p3 be Point of (TOP-REAL 2); :: thesis: cos (angle (p1,p2,p3)) = cos (angle (p3,p2,p1))

per cases ( angle (p1,p2,p3) = 0 or angle (p1,p2,p3) <> 0 ) ;

suppose angle (p1,p2,p3) = 0 ; :: thesis: cos (angle (p1,p2,p3)) = cos (angle (p3,p2,p1))

hence cos (angle (p1,p2,p3)) = cos (angle (p3,p2,p1)) by EUCLID_3:36; :: thesis: verum

end;

suppose angle (p1,p2,p3) <> 0 ; :: thesis: cos (angle (p1,p2,p3)) = cos (angle (p3,p2,p1))

then angle (p3,p2,p1) = (2 * PI) - (angle (p1,p2,p3)) by EUCLID_3:37;
then cos (angle (p1,p2,p3)) = cos ((- (angle (p3,p2,p1))) + (2 * PI))
.= cos (- (angle (p3,p2,p1))) by SIN_COS:79
.= cos (angle (p3,p2,p1)) by SIN_COS:31 ;
hence cos (angle (p1,p2,p3)) = cos (angle (p3,p2,p1)) ; :: thesis: verum

end;

end;