let X be RealNormSpace; :: thesis: for z being Element of (MetricSpaceNorm X)
for r being Real ex x being Point of X st
( x = z & cl_Ball (z,r) = { y where y is Point of X : ||.(x - y).|| <= r } )
let z be Element of (MetricSpaceNorm X); :: thesis: for r being Real ex x being Point of X st
( x = z & cl_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 & cl_Ball (z,r) = { y where y is Point of X : ||.(x - y).|| <= r } )
reconsider x = z as Point of X ;
set M = MetricSpaceNorm X;
A1: cl_Ball (z,r) = { q where q is Element of (MetricSpaceNorm X) : dist (z,q) <= r } by METRIC_1:def 15;

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 (MetricSpaceNorm X) : 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 (MetricSpaceNorm X) : 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 (MetricSpaceNorm X) : 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 (MetricSpaceNorm X) ;
||.(x - y).|| = dist (z,t) by Def1;
hence a in { q where q is Element of (MetricSpaceNorm X) : 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 (MetricSpaceNorm X) : dist (z,q) <= r } ;

now :: thesis: for a being object st a in { q where q is Element of (MetricSpaceNorm X) : 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 (MetricSpaceNorm X) : 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 (MetricSpaceNorm X) : dist (z,q) <= r } ; :: thesis: a in { y where y is Point of X : ||.(x - y).|| <= r }
then consider q being Element of (MetricSpaceNorm X) 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 (MetricSpaceNorm X) : dist (z,q) <= r } c= { y where y is Point of X : ||.(x - y).|| <= r } ;
then { q where q is Element of (MetricSpaceNorm X) : dist (z,q) <= r } = { y where y is Point of X : ||.(x - y).|| <= r } by A3, XBOOLE_0:def 10;
hence ex x being Point of X st
( x = z & cl_Ball (z,r) = { y where y is Point of X : ||.(x - y).|| <= r } ) by A1; :: thesis: verum