let A be antisymmetric RelStr ; :: thesis: for a1, a2 being Element of A st a1 <= a2 & a2 <= a1 holds
a1 = a2
let a1, a2 be Element of A; :: thesis: ( a1 <= a2 & a2 <= a1 implies a1 = a2 )
assume A1: ( [a1,a2] in the InternalRel of A & [a2,a1] in the InternalRel of A ) ; :: according to ORDERS_2:def 9 :: thesis: a1 = a2
then ( a1 in the carrier of A & a2 in the carrier of A & the InternalRel of A is_antisymmetric_in the carrier of A ) by Def6, ZFMISC_1:106;
hence a1 = a2 by A1, RELAT_2:def 4; :: thesis: verum