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

theorem
  -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,Th16;
  end;
  suppose
A3: b<0;
    --a>-b by A1,XREAL_1:24;
    then (-b)^2< a^2 by A3,Th16;
    hence thesis;
  end;
end;
