let u be Point of RealSpace ; :: thesis: for r, u1 being real number st u1 = u & r > 0 holds
Ball u,r = { s where s is Real : ( u1 - r < s & s < u1 + r ) }
let r, u1 be real number ; :: thesis: ( u1 = u & r > 0 implies Ball u,r = { s where s is Real : ( u1 - r < s & s < u1 + r ) } )
assume A1: ( u1 = u & r > 0 ) ; :: thesis: Ball u,r = { s where s is Real : ( u1 - r < s & s < u1 + r ) }
{ s where s is Real : ( u1 - r < s & s < u1 + r ) } = { q where q is Element of RealSpace : dist u,q < r }

proof

A2: { s where s is Real : ( u1 - r < s & s < u1 + r ) } c= { q where q is Element of RealSpace : dist u,q < r }

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { s where s is Real : ( u1 - r < s & s < u1 + r ) } or x in { q where q is Element of RealSpace : dist u,q < r } )
assume x in { s where s is Real : ( u1 - r < s & s < u1 + r ) } ; :: thesis: x in { q where q is Element of RealSpace : dist u,q < r }
then consider s being Real such that
A3: ( x = s & u1 - r < s & s < u1 + r ) ;
reconsider q1 = s as Point of RealSpace by METRIC_1:def 14;
(u1 - r) + r < s + r by A3, XREAL_1:8;
then A4: u1 - s < (s + r) - s by XREAL_1:11;
s - r < (u1 + r) - r by A3, XREAL_1:11;
then (s + (- r)) - s < u1 - s by XREAL_1:11;
then abs (u1 - s) < r by A4, SEQ_2:9;
then dist u,q1 < r by A1, TOPMETR:15;
hence x in { q where q is Element of RealSpace : dist u,q < r } by A3; :: thesis: verum

end;

{ q where q is Element of RealSpace : dist u,q < r } c= { s where s is Real : ( u1 - r < s & s < u1 + r ) }

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { q where q is Element of RealSpace : dist u,q < r } or x in { s where s is Real : ( u1 - r < s & s < u1 + r ) } )
assume x in { q where q is Element of RealSpace : dist u,q < r } ; :: thesis: x in { s where s is Real : ( u1 - r < s & s < u1 + r ) }
then consider q being Element of RealSpace such that
A5: ( x = q & dist u,q < r ) ;
reconsider s1 = q as Real by METRIC_1:def 14;
abs (u1 - s1) < r by A1, A5, TOPMETR:15;
then A6: ( - r < u1 - s1 & u1 - s1 < r ) by SEQ_2:9;
then (- r) + s1 < (u1 - s1) + s1 by XREAL_1:8;
then A7: (s1 - r) + r < u1 + r by XREAL_1:8;
(u1 - s1) + s1 < r + s1 by A6, XREAL_1:8;
then u1 - r < (r + s1) - r by XREAL_1:11;
hence x in { s where s is Real : ( u1 - r < s & s < u1 + r ) } by A5, A7; :: thesis: verum

end;

hence { s where s is Real : ( u1 - r < s & s < u1 + r ) } = { q where q is Element of RealSpace : dist u,q < r } by A2, XBOOLE_0:def 10; :: thesis: verum

end;

hence Ball u,r = { s where s is Real : ( u1 - r < s & s < u1 + r ) } by METRIC_1:18; :: thesis: verum