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 Th51:
  for A being Subset of TOP-REAL n st 1<=n & A is bounded holds A` <> {}
proof
  let A be Subset of TOP-REAL n;
  assume that
A1: 1<=n and
A2: A is bounded;
  consider r being Real such that
A3: for q being Point of TOP-REAL n st q in A holds |.q.|<r by A2,Th21;
  |.r.|>=0 by COMPLEX1:46;
  then
A4: |.r.|*|.1.REAL n.|>=|.r.|*1 by A1,EUCLID:75,XREAL_1:64;
  |.r*(1.REAL n).|=|.r.|*|.1.REAL n.| & r<=|.r.| by ABSVALUE:4,TOPRNS_1:7;
  then not r*(1.REAL n) in A by A3,A4,XXREAL_0:2;
  hence thesis by XBOOLE_0:def 5;
end;
