let X be RealNormSpace; :: thesis: for z being Element of ()
for r being Real ex x being Point of X st
( x = z & Ball (z,r) = { y where y is Point of X : ||.(x - y).|| < r } )

let z be Element of (); :: thesis: for r being Real ex x being Point of X st
( x = z & Ball (z,r) = { y where y is Point of X : ||.(x - y).|| < r } )

let r be Real; :: thesis: ex x being Point of X st
( x = z & Ball (z,r) = { y where y is Point of X : ||.(x - y).|| < r } )

set M = MetricSpaceNorm X;
reconsider x = z as Point of X ;
A1: Ball (z,r) = { q where q is Element of () : dist (z,q) < r } by METRIC_1:def 14;
now :: thesis: for a being object st a in { y where y is Point of X : ||.(x - y).|| < r } holds
a in { q where q is Element of () : dist (z,q) < r }
let a be object ; :: thesis: ( a in { y where y is Point of X : ||.(x - y).|| < r } implies a in { q where q is Element of () : dist (z,q) < r } )
assume a in { y where y is Point of X : ||.(x - y).|| < r } ; :: thesis: a in { q where q is Element of () : dist (z,q) < r }
then consider y being Point of X such that
A2: ( a = y & ||.(x - y).|| < r ) ;
reconsider t = y as Element of () ;
||.(x - y).|| = dist (z,t) by Def1;
hence a in { q where q is Element of () : dist (z,q) < r } by A2; :: thesis: verum
end;
then A3: { y where y is Point of X : ||.(x - y).|| < r } c= { q where q is Element of () : dist (z,q) < r } ;
now :: thesis: for a being object st a in { q where q is Element of () : dist (z,q) < r } holds
a in { y where y is Point of X : ||.(x - y).|| < r }
let a be object ; :: thesis: ( a in { q where q is Element of () : dist (z,q) < r } implies a in { y where y is Point of X : ||.(x - y).|| < r } )
assume a in { q where q is Element of () : dist (z,q) < r } ; :: thesis: a in { y where y is Point of X : ||.(x - y).|| < r }
then consider q being Element of () such that
A4: ( a = q & dist (z,q) < r ) ;
reconsider t = q as Point of X ;
||.(x - t).|| = dist (z,q) by Def1;
hence a in { y where y is Point of X : ||.(x - y).|| < r } by A4; :: thesis: verum
end;
then { q where q is Element of () : dist (z,q) < r } c= { y where y is Point of X : ||.(x - y).|| < r } ;
then { q where q is Element of () : dist (z,q) < r } = { y where y is Point of X : ||.(x - y).|| < r } by ;
hence ex x being Point of X st
( x = z & Ball (z,r) = { y where y is Point of X : ||.(x - y).|| < r } ) by A1; :: thesis: verum