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Analyses and syntheses are a pair of fundamental cognitive processes at the meta-inference layer of the brain. This paper presents the cognitive processes of analysis and synthesis as a part of the brainpsilas inference mechanisms. The cognitive foundations of analysis and synthesis are explored, and their mathematical models are created in concept algebra and system algebra by concept and/or system decomposition, composition, specification, and aggregation. Based on the cognitive and mathematical models, the cognitive processes of analysis and synthesis are formally described in real-time process algebra (RTPA). The formal modeling and rigorous explanation of this pair of cognitive processes for formal inference reveal a part of the mechanisms of the brain as modeled in the layered reference model of the brain (LRMB).