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

proof

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 set ; :: 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 ;
reconsider eu = u, eq = q as Element of REAL 1 ;
q is Tuple of 1, REAL by FINSEQ_2:151;
then consider s1 being Real such that
A5: q = <*s1*> by FINSEQ_2:117;
<*u1*> - <*s1*> = <*(u1 - s1)*> by RVSUM_1:50;
then sqr (<*u1*> - <*s1*>) = <*((u1 - s1) ^2)*> by RVSUM_1:81;
then Sum (sqr (<*u1*> - <*s1*>)) = (u1 - s1) ^2 by FINSOP_1:12;
then A6: sqrt (Sum (sqr (<*u1*> - <*s1*>))) = abs (u1 - s1) by COMPLEX1:158;
(Pitag_dist 1) . (eu,eq) < r by A4, METRIC_1:def 1;
then A7: |.(<*u1*> - <*s1*>).| < r by A1, A5, EUCLID:def 6;
then u1 - s1 < r by A6, SEQ_2:9;
then (u1 - s1) + s1 < r + s1 by XREAL_1:8;
then A8: u1 - r < (r + s1) - r by XREAL_1:11;
- r < u1 - s1 by A7, A6, SEQ_2:9;
then (- r) + s1 < (u1 - s1) + s1 by XREAL_1:8;
then (s1 - r) + r < u1 + r by XREAL_1:8;
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_2:118;
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 (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_1:11;
then A12: (s + (- r)) - s < u1 - s by XREAL_1:11;
reconsider es = <*s*> as Element of REAL 1 by FINSEQ_2:118;
reconsider q1 = <*s*> as Element of (Euclid 1) by FINSEQ_2:118;
<*u1*> - <*s*> = <*(u1 - s)*> by RVSUM_1:50;
then sqr (<*u1*> - <*s*>) = <*((u1 - s) ^2)*> by RVSUM_1:81;
then A13: Sum (sqr (<*u1*> - <*s*>)) = (u1 - s) ^2 by FINSOP_1:12;
(u1 - r) + r < s + r by A10, XREAL_1:8;
then u1 - s < (s + r) - s by XREAL_1:11;
then abs (u1 - s) < r by A12, SEQ_2:9;
then |.(<*u1*> - <*s*>).| < r by A13, COMPLEX1:158;
then ( the distance of (Euclid 1) . (u,q1) = dist (u,q1) & (Pitag_dist 1) . (eu1,es) < r ) by EUCLID:def 6, METRIC_1:def 1;
hence x in { q where q is Element of (Euclid 1) : dist (u,q) < r } by A1, A9; :: thesis: verum

end;

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