let R be Relation; for X being set st R linearly_orders X holds
R ~ linearly_orders X
let X be set ; ( R linearly_orders X implies R ~ linearly_orders X )
assume that
A1:
R is_reflexive_in X
and
A2:
R is_transitive_in X
and
A3:
R is_antisymmetric_in X
and
A4:
R is_connected_in X
; ORDERS_1:def 8 R ~ linearly_orders X
thus
( R ~ is_reflexive_in X & R ~ is_transitive_in X & R ~ is_antisymmetric_in X & R ~ is_connected_in X )
by A1, A2, A3, A4, Lm11, Lm12, Lm13, Lm14; ORDERS_1:def 8 verum