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 Th37:
  for a being Real st n>=1 holds {q: |.q.| >a} <>{}
proof
  let a be Real;
  assume
A1: n>=1;
  now
    |.a+1.|>=0 & sqrt 1<=sqrt n by A1,COMPLEX1:46,SQUARE_1:26;
    then
A2: |.a+1.|*1<=|.a+1.|*sqrt n by XREAL_1:64;
A3: a+1<=|.a+1.| by ABSVALUE:4;
    assume not (a+1)*(1.REAL n) in {q : |.q.| > a };
    then
A4: |.(a+1)*(1.REAL n).|<=a;
A5: a<a+1 by XREAL_1:29;
    |.(a+1)*(1.REAL n).|=|.a+1.|*|.(1.REAL n).| by TOPRNS_1:7
      .=|.a+1.|*(sqrt n) by EUCLID:73;
    then a+1<= |.(a+1)*(1.REAL n).| by A2,A3,XXREAL_0:2;
    hence contradiction by A4,A5,XXREAL_0:2;
  end;
  hence thesis;
end;
