let R be Relation; :: thesis: ( R is being_quasi-order implies R ~ is being_quasi-order )
assume that
A1: R is reflexive and
A2: R is transitive ; :: according to ORDERS_1:def 3 :: thesis: R ~ is being_quasi-order
thus ( R ~ is reflexive & R ~ is transitive ) by A1, A2, RELAT_2:27, RELAT_2:42; :: according to ORDERS_1:def 3 :: thesis: verum