let P, Q, R be Relation; :: thesis: ( P c= Q implies P * R c= Q * R )
assume A1: P c= Q ; :: thesis: P * R c= Q * R
let x be set ; :: according to RELAT_1:def 3 :: thesis: for b being set st [x,b] in P * R holds
[x,b] in Q * R
let y be set ; :: thesis: ( [x,y] in P * R implies [x,y] in Q * R )
assume [x,y] in P * R ; :: thesis: [x,y] in Q * R
then ex z being set st
( [x,z] in P & [z,y] in R ) by Def8;
hence [x,y] in Q * R by A1, Def8; :: thesis: verum