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

proof

assume w in Ball x,r ; :: thesis: dist x,w < r
then ||.(x - w).|| < r by Th40;
hence dist x,w < r by CSSPACE:def 16; :: thesis: verum

end;

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