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