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
  for x, a being Real st |.x.|<=a holds x^2<=a^2
proof
  let x, a be Real;
  assume
A1: |.x.|<=a;
  per cases;
  suppose
A2: x>=0;
    x<=a by A1,ABSVALUE:def 1;
    hence thesis by A2,SQUARE_1:15;
  end;
  suppose
A3: x<0;
    then -x<=a by A1,ABSVALUE:def 1;
    then (-x)^2<=a^2 by A3,SQUARE_1:15;
    hence thesis;
  end;
end;
