α∥β ⟺ α∩β=∅\alpha \parallel \beta \iff \alpha\cap\beta=\varnothingα∥β⟺α∩β=∅.
α∥β, a=γ∩α, b=γ∩β ⟹ a∥b\alpha\parallel \beta,~a=\gamma\cap \alpha,~b=\gamma\cap\beta \implies a\parallel bα∥β, a=γ∩α, b=γ∩β⟹a∥b.
α∥β ⟸ a1,a2⊂α, a1∩a2≠∅, b1,b2⊂β, b1∩b2≠∅, a1∥b1, a2∥b2\alpha\parallel \beta \impliedby a_1,a_2\subset \alpha,~a_1\cap a_2\ne\varnothing,~b_1,b_2\subset \beta,~b_1\cap b_2\ne\varnothing,~a_1\parallel b_1,~a_2\parallel b_2α∥β⟸a1,a2⊂α, a1∩a2=∅, b1,b2⊂β, b1∩b2=∅, a1∥b1, a2∥b2.
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