let X be ComplexUnitarySpace; :: thesis: for x, w being Point of X
for r being Real holds
( w in cl_Ball (x,r) iff dist (x,w) <= r )
let x, w be Point of X; :: thesis: for r being Real holds
( w in cl_Ball (x,r) iff dist (x,w) <= r )
let r be Real; :: thesis: ( w in cl_Ball (x,r) iff dist (x,w) <= r )
thus ( w in cl_Ball (x,r) implies dist (x,w) <= r ) :: thesis: ( dist (x,w) <= r implies w in cl_Ball (x,r) )

proof

assume w in cl_Ball (x,r) ; :: thesis: dist (x,w) <= r
then ||.(x - w).|| <= r by Th47;
hence dist (x,w) <= r by CSSPACE:def 16; :: thesis: verum

end;

assume dist (x,w) <= r ; :: thesis: w in cl_Ball (x,r)
then ||.(x - w).|| <= r by CSSPACE:def 16;
hence w in cl_Ball (x,r) ; :: thesis: verum