reserve a,b,c,x,y,z for Real;

theorem Th49:
  -a <= b & b <= a implies b^2 <= a^2
proof
  assume that
A1: -a<=b and
A2: b<=a;
  per cases;
  suppose
    b>=0;
    hence thesis by A2,Th15;
  end;
  suppose
A3: b<0;
    --a>=-b by A1,XREAL_1:24;
    then (-b)^2<= a^2 by A3,Th15;
    hence thesis;
  end;
end;
