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 Th20:
  for W being Subset of Euclid n st n>=1 & W=REAL n holds W is not bounded
proof
  let W be Subset of Euclid n;
  assume that
A1: n>=1 and
A2: W=(REAL n);
  reconsider y0=0.TOP-REAL n as Point of Euclid n by EUCLID:67;
  assume W is bounded;
  then consider r being Real such that
A3: 0<r and
A4: for x,y being Point of Euclid n st x in W & y in W holds dist(x,y)<=
  r;
  reconsider x0=(r+1)*(1.REAL n) as Point of Euclid n by TOPREAL3:8;
  dist(x0,y0)<=r by A2,A4;
  then |.(r+1)*(1.REAL n) -0.TOP-REAL n.|<=r by JGRAPH_1:28;
  then |.(r+1)*(1.REAL n).|<=r by RLVECT_1:13;
  then |.r+1.|*|.(1.REAL n).|<=r by TOPRNS_1:7;
  then |.r+1.|*(sqrt n)<=r by EUCLID:73;
  then
A5: (r+1)*(sqrt n)<=r by A3,ABSVALUE:def 1;
  (sqrt 1)<=(sqrt n) by A1,SQUARE_1:26;
  then (r+1)*1<=(r+1)*(sqrt n) by A3,XREAL_1:64;
  then (r+1)*1<=r by A5,XXREAL_0:2;
  then (r+1)-r<=r-r by XREAL_1:9;
  then 1<=0;
  hence contradiction;
end;
