set X = the carrier of R \/ the carrier of S;

A1: the carrier of S c= the carrier of R \/ the carrier of S by XBOOLE_1:7;

the carrier of R 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 A1, RELSET_1:7;

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

A1: the carrier of S c= the carrier of R \/ the carrier of S by XBOOLE_1:7;

the carrier of R 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 A1, RELSET_1:7;

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