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,Th15;
        hence contradiction by A1;
      end;
      case
        b>=a;
        hence contradiction by A1,A2,Th15;
      end;
    end;
    hence contradiction;
  end;
  hence thesis;
end;
