let X be RealUnitarySpace; :: thesis: for x, z being Point of X

for r being Real holds

( z in Ball (x,r) iff ||.(x - z).|| < r )

let x, z be Point of X; :: thesis: for r being Real holds

( z in Ball (x,r) iff ||.(x - z).|| < r )

let r be Real; :: thesis: ( z in Ball (x,r) iff ||.(x - z).|| < r )

thus ( z in Ball (x,r) implies ||.(x - z).|| < r ) :: thesis: ( ||.(x - z).|| < r implies z in Ball (x,r) )

for r being Real holds

( z in Ball (x,r) iff ||.(x - z).|| < r )

let x, z be Point of X; :: thesis: for r being Real holds

( z in Ball (x,r) iff ||.(x - z).|| < r )

let r be Real; :: thesis: ( z in Ball (x,r) iff ||.(x - z).|| < r )

thus ( z in Ball (x,r) implies ||.(x - z).|| < r ) :: thesis: ( ||.(x - z).|| < r implies z in Ball (x,r) )

proof

thus
( ||.(x - z).|| < r implies z in Ball (x,r) )
; :: thesis: verum
assume
z in Ball (x,r)
; :: thesis: ||.(x - z).|| < r

then ex y being Point of X st

( z = y & ||.(x - y).|| < r ) ;

hence ||.(x - z).|| < r ; :: thesis: verum

end;then ex y being Point of X st

( z = y & ||.(x - y).|| < r ) ;

hence ||.(x - z).|| < r ; :: thesis: verum