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_3:45; :: 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_3:46;
then cos (angle p1,p2,p3) = cos ((- (angle p3,p2,p1)) + (2 * PI ))
.= cos (- (angle p3,p2,p1)) by SIN_COS:84
.= cos (angle p3,p2,p1) by SIN_COS:34 ;
hence cos (angle p1,p2,p3) = cos (angle p3,p2,p1) ; :: thesis: verum

end;

end;