let P, R be Relation; :: thesis: ( P is transitive & R is transitive implies P /\ R is transitive )
assume that
A1: P is transitive and
A2: R is transitive ; :: thesis: P /\ R is transitive
A3: R is_transitive_in field R by A2, Def16;
A4: P is_transitive_in field P by A1, Def16;

now

let a, b, c be set ; :: thesis: ( a in field (P /\ R) & b in field (P /\ R) & c in field (P /\ R) & [a,b] in P /\ R & [b,c] in P /\ R implies [a,c] in P /\ R )
assume that
a in field (P /\ R) and
b in field (P /\ R) and
c in field (P /\ R) and
A5: [a,b] in P /\ R and
A6: [b,c] in P /\ R ; :: thesis: [a,c] in P /\ R
A7: [b,c] in R by A6, XBOOLE_0:def 4;
then A8: c in field R by RELAT_1:15;
A9: [a,b] in R by A5, XBOOLE_0:def 4;
then ( a in field R & b in field R ) by RELAT_1:15;
then A10: [a,c] in R by A3, A9, A7, A8, Def8;
A11: [b,c] in P by A6, XBOOLE_0:def 4;
then A12: c in field P by RELAT_1:15;
A13: [a,b] in P by A5, XBOOLE_0:def 4;
then ( a in field P & b in field P ) by RELAT_1:15;
then [a,c] in P by A4, A13, A11, A12, Def8;
hence [a,c] in P /\ R by A10, XBOOLE_0:def 4; :: thesis: verum

end;

then P /\ R is_transitive_in field (P /\ R) by Def8;
hence P /\ R is transitive by Def16; :: thesis: verum