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 Th18:
  <*u1*> = U & r > 0 implies Fr Ball(U,r) = {<*u1-r*>,<*u1+r*>}
proof
  assume that
A1: <*u1*>=U and
A2: r>0;
  reconsider u=U as Point of Euclid 1 by Lm6;
A3: Ball(u,r)={<*s*>:u1-r<s & s<u1+r} by A1,JORDAN2B:27;
  Ball(U,r)=Ball(u,r) by TOPREAL9:13;
  then Ball(U,r) is open by KURATO_2:1;
  then
A4: Fr Ball(U,r)=Cl Ball(U,r)\Ball(U,r) by TOPS_1:42
    .=cl_Ball(U,r)\Ball(U,r) by A2,JORDAN:23
    .=cl_Ball(u,r)\Ball(U,r) by TOPREAL9:14
    .=cl_Ball(u,r)\Ball(u,r) by TOPREAL9:13;
A5: cl_Ball(u,r)={<*s*>:u1-r<=s & s<=u1+r} by A1,Th17;
  thus Fr Ball(U,r)c={<*u1-r*>,<*u1+r*>}
  proof
    let x be object such that
A6: x in Fr Ball(U,r);
    reconsider X=x as Point of Euclid 1 by Lm6,A6;
    x in cl_Ball(u,r) by A4,A6,XBOOLE_0:def 5;
    then consider s be Real such that
A7: x=<*s*> and
A8: u1-r<=s and
A9: s<=u1+r by A5;
    assume
A10: not x in {<*u1-r*>,<*u1+r*>};
    then s<>u1+r by A7,TARSKI:def 2;
    then
A11: s<u1+r by A9,XXREAL_0:1;
    s<>u1-r by A7,A10,TARSKI:def 2;
    then u1-r<s by A8,XXREAL_0:1;
    then X in Ball(u,r) by A3,A7,A11;
    hence thesis by A4,A6,XBOOLE_0:def 5;
  end;
  let x be object;
  assume x in {<*u1-r*>,<*u1+r*>};
  then
A12: x=<*u1-r*> or x=<*u1+r*> by TARSKI:def 2;
  assume
A13: not x in Fr Ball(U,r);
  u1+-r<=u1+r by A2,XREAL_1:6;
  then x in cl_Ball(u,r) by A5,A12;
  then x in Ball(u,r) by A4,A13,XBOOLE_0:def 5;
  then ex s be Real st x=<*s*> & u1-r<s & s<u1+r by A3;
  hence contradiction by A12,FINSEQ_1:76;
end;
