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Equality Reasoning

forall A, B, C: (B union A) intersection (C union A) = (B intersection C) union A.

Venn diagram:

We prove the proposition as follows:

(B union A) intersection (C union A) = (commutativity)

(A union B) intersection (A union C) = (distributivity)

A union (B intersection C) = (commutativity)

(B intersection C) union A.

Author: Wolfgang Schreiner
Last Modification: October 14, 1999

(`B` union `A`) intersection (`C` union `A`)	=	(commutativity)
(`A` union `B`) intersection (`A` union `C`)	=	(distributivity)
`A` union (`B` intersection `C`)	=	(commutativity)
(`B` intersection `C`) union `A`.