theorem
  F is associative implies F[:](f, F.(x1,x2)) = F[:](F[:](f,x1),x2)
proof
  assume
A1: F is associative;
  per cases;
  suppose
    Y = {};
    hence thesis;
  end;
  suppose
A2: Y <> {};
    now
      let y;
      reconsider x3 = f.y as Element of X by A2,FUNCT_2:5;
      thus (F[:](f, F.(x1,x2))).y = F.(f.y, F.(x1,x2)) by A2,Th48
        .= F.(F.(x3,x1),x2) by A1
        .= F.(F[:](f,x1).y,x2) by A2,Th48;
    end;
    hence thesis by A2,Th49;
  end;
end;
