A1: R is_asymmetric_in field R by Def13;
A2: field (P /\ R) c= (field P) /\ (field R) by RELAT_1:19;
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
A3: ( a in field (P /\ R) & b in field (P /\ R) ) and
A4: [a,b] in P /\ R ; :: thesis: not [b,a] in P /\ R
A5: [a,b] in R by A4, XBOOLE_0:def 4;
( a in field R & b in field R ) by A2, A3, XBOOLE_0:def 4;
then not [b,a] in R by A1, A5;
hence not [b,a] in P /\ R by XBOOLE_0:def 4; :: thesis: verum