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In Q, Jesus is lined up with the prophets, and in the pattern of Deuteronomy, and he is rejected by the people. By the time Mark is written, the passion has been added.
Q is defined as that material found in both Matthew and Luke but not found in Mark. But if Matthew declined to use some passages found in his copy of Q because it didn't fit his agenda (Jesus as the new Moses), then at least part of the L source was found in the lost Q. And the reverse is true for the M source (Luke had an anti-Galilean bent). The underlying assumption is that each of the gospel authors used every scrap of everything he had on hand.