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

theorem
  b^2 <= a^2 & a >= 0 implies -a <= b & b <= a
proof
  assume that
A1: b^2<= a^2 and
A2: a>=0;
  now
    assume
A3: -a>b or b>a;
    now
      per cases by A3;
      case
        -a>b;
        then --a<-b by XREAL_1:24;
        then a^2<(-b)^2 by A2,Th16;
        hence contradiction by A1;
      end;
      case
        b>a;
        hence contradiction by A1,A2,Th16;
      end;
    end;
    hence contradiction;
  end;
  hence thesis;
end;
