let A be transitive RelStr ; :: thesis: for a1, a2, a3 being Element of A st a1 <= a2 & a2 <= a3 holds
a1 <= a3
let a1, a2, a3 be Element of A; :: thesis: ( a1 <= a2 & a2 <= a3 implies a1 <= a3 )
assume that
A1: [a1,a2] in the InternalRel of A and
A2: [a2,a3] in the InternalRel of A ; :: according to ORDERS_2:def 5 :: thesis: a1 <= a3
A3: the InternalRel of A is_transitive_in the carrier of A by Def3;
a1 in the carrier of A by A1, ZFMISC_1:87;
hence [a1,a3] in the InternalRel of A by A1, A2, A3; :: according to ORDERS_2:def 5 :: thesis: verum