reserve n for Nat,
        X for set,
        Fx,Gx for Subset-Family of X;
reserve TM for metrizable TopSpace,
        TM1,TM2 for finite-ind second-countable metrizable TopSpace,
        A,B,L,H for Subset of TM,
        U,W for open Subset of TM,
        p for Point of TM,

        F,G for finite Subset-Family of TM,
        I for Integer;
reserve u for Point of Euclid 1,
  U for Point of TOP-REAL 1,
  r,u1 for Real,
  s for Real;

theorem Th17:
  <*u1*> = u implies cl_Ball(u,r)={<*s*>:u1-r <= s & s <= u1+r}
proof
  assume that
A1: <*u1*>=u;
  set E1={q where q is Element of Euclid 1:dist(u,q)<=r};
  reconsider u1 as Element of REAL by XREAL_0:def 1;
  set R1={<*s*> where s is Real:u1-r<=s & s<=u1+r};
A2: E1 c=R1
  proof
    let x be object;
    assume x in E1;
    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 be 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 thesis by A3,A5,A8;
  end;
  R1 c=E1
  proof
    reconsider eu1=<*u1*> as Element of REAL 1 by FINSEQ_2:98;
    let x be object;
    assume x in R1;
    then consider s be 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 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 thesis by A1,A9;
  end;
  then E1=R1 by A2;
  hence thesis by METRIC_1:18;
end;
