let R be Relation; :: thesis: for X being set st R partially_orders X holds
R |_2 X is being_partial-order
let X be set ; :: thesis: ( R partially_orders X implies R |_2 X is being_partial-order )
assume that
A1: R is_reflexive_in X and
A2: R is_transitive_in X and
A3: R is_antisymmetric_in X ; :: according to ORDERS_1:def 7 :: thesis: R |_2 X is being_partial-order
thus ( R |_2 X is reflexive & R |_2 X is transitive & R |_2 X is antisymmetric ) by A1, A2, A3, Lm6, Lm7, Lm8; :: according to ORDERS_1:def 4 :: thesis: verum