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When you Special Summon "Theinen the Great Sphinx" you do not have to pay the additional 500 Life Points to increase its ATK, you can choose to leave the ATK at 3500. If you do not pay the extra 500 when it is Summoned, you cannot later decide to pay and increase its ATK, it must be done when it is Special Summoned.
The only way you can Special Summon this card is via its trigger effect. Even if it was initially summoned that way, you still can't summon it with card effects.
The effect of Summoning "Theinen the Great Sphinx" is a Trigger Effect, so you may do it during your opponent's turn if "Andro Sphinx" and "Sphinx Teleia" are destroyed simultaneously.
The effect to summon this card is a Trigger effect. It can activate while in the hand or Deck. You can't activate the effects of 2 or more of this card's effect in the same Chain, so you can only Summon 1 "Thenien the Great Sphinx" at a time.
Since Summoning "Theinen the Great Sphinx" is a Spell Speed 1 effect (Trigger Effect), you cannot chain multiple Special Summons of "Theinen the Great Sphinx" to one another. This means you can only Special Summon 1 copy of "Theinen the Great Sphinx" when the conditions are correct. You CANNOT pay 1000 Life Points to Special Summon 2 copies of this card.
Both of this card's effects are optional, and both can can miss their timing.