set X = the carrier of R \/ the carrier of S;
( the carrier of R c= the carrier of R \/ the carrier of S & the carrier of S c= the carrier of R \/ the carrier of S ) by XBOOLE_1:7;
then reconsider IR = the InternalRel of R, IS = the InternalRel of S as Relation of (the carrier of R \/ the carrier of S) by RELSET_1:17;
set D = IR \/ IS;
reconsider D = IR \/ IS as Relation of (the carrier of R \/ the carrier of S) ;
take RelStr(# (the carrier of R \/ the carrier of S),D #) ; :: thesis: ( the carrier of RelStr(# (the carrier of R \/ the carrier of S),D #) = the carrier of R \/ the carrier of S & the InternalRel of RelStr(# (the carrier of R \/ the carrier of S),D #) = the InternalRel of R \/ the InternalRel of S )
thus ( the carrier of RelStr(# (the carrier of R \/ the carrier of S),D #) = the carrier of R \/ the carrier of S & the InternalRel of RelStr(# (the carrier of R \/ the carrier of S),D #) = the InternalRel of R \/ the InternalRel of S ) ; :: thesis: verum