reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;

theorem Th23:
  n>=1 implies UBD {}(TOP-REAL n)=REAL n
proof
  set A={}(TOP-REAL n);
A1: (TOP-REAL n) | [#](TOP-REAL n)=the TopStruct of TOP-REAL n by TSEP_1:93;
  assume
A2: n>=1;
A3: now
    reconsider D1=[#]((TOP-REAL n) | A`) as Subset of Euclid n
    by A1,TOPREAL3:8;
    assume for D being Subset of Euclid n st D=[#]((TOP-REAL n) | A`) holds D
    is bounded;
    then D1 is bounded;
    then [#](TOP-REAL n) is bounded by A1,Th5;
    hence contradiction by A2,Th22;
  end;
  [#]((TOP-REAL n) | A`) is a_component by A1,CONNSP_1:45;
  then [#](TOP-REAL n) is_outside_component_of {}(TOP-REAL n) by A1,A3,Th8;
  then
A4: [#](TOP-REAL n) in {B2 where B2 is Subset of TOP-REAL n: B2
  is_outside_component_of {}(TOP-REAL n)};
  UBD {}(TOP-REAL n) c= the carrier of TOP-REAL n;
  hence UBD {}(TOP-REAL n) c= REAL n by EUCLID:22;
  let x be object;
  assume x in REAL n;
  then x in [#](TOP-REAL n) by EUCLID:22;
  hence thesis by A4,TARSKI:def 4;
end;
