reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  |.x.|>=a implies x^2>=a^2
proof
  assume
A1: |.x.|>=a;
  per cases;
  suppose
    x>=0;
    then x>=a by A1,ABSVALUE:def 1;
    hence thesis by SQUARE_1:15;
  end;
  suppose
    x<0;
    then -x>=a by A1,ABSVALUE:def 1;
    then (-x)^2>=a^2 by SQUARE_1:15;
    hence thesis;
  end;
end;
