A7: R is_symmetric_in field R by Def11;
A8: P is_symmetric_in field P by Def11;

now :: thesis: for a, b being object st a in field (P /\ R) & b in field (P /\ R) & [a,b] in P /\ R holds
[b,a] in P /\ R

let a, b be object ; :: thesis: ( a in field (P /\ R) & b in field (P /\ R) & [a,b] in P /\ R implies [b,a] in P /\ R )
assume that
A9: ( a in field (P /\ R) & b in field (P /\ R) ) and
A10: [a,b] in P /\ R ; :: thesis: [b,a] in P /\ R
A11: [a,b] in R by A10, XBOOLE_0:def 4;
A12: field (P /\ R) c= (field P) /\ (field R) by RELAT_1:19;
then ( a in field R & b in field R ) by A9, XBOOLE_0:def 4;
then A13: [b,a] in R by A7, A11;
A14: [a,b] in P by A10, XBOOLE_0:def 4;
( a in field P & b in field P ) by A12, A9, XBOOLE_0:def 4;
then [b,a] in P by A8, A14;
hence [b,a] in P /\ R by A13, XBOOLE_0:def 4; :: thesis: verum

end;

hence P /\ R is symmetric by Def3; :: thesis: verum