let u be Point of (Euclid 1); :: thesis: for r, u1 being Real st <*u1*> = u holds
cl_Ball (u,r) = { <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) }
let r, u1 be Real; :: thesis: ( <*u1*> = u implies cl_Ball (u,r) = { <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) } )
assume A1: <*u1*> = u ; :: thesis: cl_Ball (u,r) = { <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) }
set E1 = { q where q is Element of (Euclid 1) : dist (u,q) <= r } ;
reconsider u1 = u1 as Element of REAL by XREAL_0:def 1;
set R1 = { <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) } ;
A2: { q where q is Element of (Euclid 1) : dist (u,q) <= r } c= { <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) }

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { q where q is Element of (Euclid 1) : 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 (Euclid 1) : dist (u,q) <= r } ; :: thesis: x in { <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) }
then consider q being Element of (Euclid 1) such that
A3: x = q and
A4: dist (u,q) <= r ;
q is Tuple of 1, REAL by FINSEQ_2:131;
then consider s1 being Element of REAL such that
A5: q = <*s1*> by FINSEQ_2:97;
<*u1*> - <*s1*> = <*(u1 - s1)*> by RVSUM_1:29;
then sqr (<*u1*> - <*s1*>) = <*((u1 - s1) ^2)*> by RVSUM_1:55;
then Sum (sqr (<*u1*> - <*s1*>)) = (u1 - s1) ^2 by RVSUM_1:73;
then A6: sqrt (Sum (sqr (<*u1*> - <*s1*>))) = |.(u1 - s1).| by COMPLEX1:72;
A7: |.(<*u1*> - <*s1*>).| <= r by A1, A4, A5, EUCLID:def 6;
then u1 - s1 <= r by A6, ABSVALUE:5;
then (u1 - s1) + s1 <= r + s1 by XREAL_1:6;
then A8: u1 - r <= (r + s1) - r by XREAL_1:9;
- r <= u1 - s1 by A6, A7, ABSVALUE:5;
then (- r) + s1 <= (u1 - s1) + s1 by XREAL_1:6;
then (s1 - r) + r <= u1 + r by XREAL_1:6;
hence x in { <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) } by A3, A5, A8; :: thesis: verum

end;

{ <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) } c= { q where q is Element of (Euclid 1) : dist (u,q) <= r }

proof

reconsider eu1 = <*u1*> as Element of REAL 1 by FINSEQ_2:98;
let x be object ; :: 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 (Euclid 1) : 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 (Euclid 1) : dist (u,q) <= r }
then consider s being Real such that
A9: x = <*s*> and
A10: u1 - r <= s and
A11: s <= u1 + r ;
s - r <= (u1 + r) - r by A11, XREAL_1:9;
then A12: (s + (- r)) - s <= u1 - s by XREAL_1:9;
reconsider s = s as Element of REAL by XREAL_0:def 1;
reconsider es = <*s*> as Element of REAL 1 by FINSEQ_2:98;
reconsider q1 = <*s*> as Element of (Euclid 1) by FINSEQ_2:98;
<*u1*> - <*s*> = <*(u1 - s)*> by RVSUM_1:29;
then sqr (<*u1*> - <*s*>) = <*((u1 - s) ^2)*> by RVSUM_1:55;
then A13: Sum (sqr (<*u1*> - <*s*>)) = (u1 - s) ^2 by RVSUM_1:73;
(u1 - r) + r <= s + r by A10, XREAL_1:6;
then u1 - s <= (s + r) - s by XREAL_1:9;
then |.(u1 - s).| <= r by A12, ABSVALUE:5;
then |.(<*u1*> - <*s*>).| <= r by A13, COMPLEX1:72;
then ( the distance of (Euclid 1) . (u,q1) = dist (u,q1) & (Pitag_dist 1) . (eu1,es) <= r ) by EUCLID:def 6;
hence x in { q where q is Element of (Euclid 1) : dist (u,q) <= r } by A1, A9; :: thesis: verum

end;

then { q where q is Element of (Euclid 1) : dist (u,q) <= r } = { <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) } by A2;
hence cl_Ball (u,r) = { <*s*> where s is Real : ( u1 - r <= s & s <= u1 + r ) } by METRIC_1:18; :: thesis: verum