let R be Relation; for X, Y being set st R partially_orders Y & X c= Y holds
R partially_orders X
let X, Y be set ; ( R partially_orders Y & X c= Y implies R partially_orders X )
assume that
A1:
R is_reflexive_in Y
and
A2:
R is_transitive_in Y
and
A3:
R is_antisymmetric_in Y
and
A4:
X c= Y
; ORDERS_1:def 8 R partially_orders X
thus
( R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X )
by A1, A2, A3, A4; ORDERS_1:def 8 verum