A1: R is_symmetric_in field R by Def11;
A2: 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
a in field (P \/ R) and
b in field (P \/ R) and
A3: [a,b] in P \/ R ; :: thesis: [b,a] in P \/ R

A4: now :: thesis: ( [a,b] in R implies [b,a] in P \/ R )

assume A5: [a,b] in R ; :: thesis: [b,a] in P \/ R
then ( a in field R & b in field R ) by RELAT_1:15;
then [b,a] in R by A1, A5;
hence [b,a] in P \/ R by XBOOLE_0:def 3; :: thesis: verum

end;

now :: thesis: ( [a,b] in P implies [b,a] in P \/ R )

assume A6: [a,b] in P ; :: thesis: [b,a] in P \/ R
then ( a in field P & b in field P ) by RELAT_1:15;
then [b,a] in P by A2, A6;
hence [b,a] in P \/ R by XBOOLE_0:def 3; :: thesis: verum

end;

hence [b,a] in P \/ R by A3, A4, XBOOLE_0:def 3; :: thesis: verum

end;

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