let R be Relation; :: thesis: ( R is well-ordering implies ( R is being_quasi-order & R is being_partial-order & R is being_linear-order ) )
assume that
A1: R is reflexive and
A2: R is transitive and
A3: R is antisymmetric and
A4: R is connected and
R is well_founded ; :: according to WELLORD1:def 4 :: thesis: ( R is being_quasi-order & R is being_partial-order & R is being_linear-order )
thus ( R is reflexive & R is transitive & R is reflexive & R is transitive & R is antisymmetric & R is reflexive & R is transitive & R is antisymmetric & R is connected ) by A1, A2, A3, A4; :: according to ORDERS_1:def 3,ORDERS_1:def 4,ORDERS_1:def 5 :: thesis: verum