A1: P is_asymmetric_in field P by Def13;
let a be object ; :: according to RELAT_2:def 5,RELAT_2:def 13 :: thesis: for y being object st a in field (P \ R) & y in field (P \ R) & [a,y] in P \ R holds
not [y,a] in P \ R
let b be object ; :: thesis: ( a in field (P \ R) & b in field (P \ R) & [a,b] in P \ R implies not [b,a] in P \ R )
assume that
a in field (P \ R) and
b in field (P \ R) and
A2: [a,b] in P \ R ; :: thesis: not [b,a] in P \ R
A3: [a,b] in P by A2, XBOOLE_0:def 5;
then ( a in field P & b in field P ) by RELAT_1:15;
then not [b,a] in P by A1, A3;
hence not [b,a] in P \ R by XBOOLE_0:def 5; :: thesis: verum