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We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the $omega_1$-approximation property. We prove that the existence of stationarily many $omega_1$-guessing models in $P_{omega_2}(H(theta))$, for sufficiently large cardinals $theta$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss.

QUOTIENTS OF STRONGLY PROPER FORCINGS AND GUESSING MODELS