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