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;
A2:
the carrier of R c= the carrier of R \/ the carrier of S
by XBOOLE_1:7;
then reconsider IQ = [: the carrier of R, the carrier of S:] as Relation of ( the carrier of R \/ the carrier of S) by A1, ZFMISC_1:96;
reconsider IP = [: the carrier of S, the carrier of R:] as Relation of ( the carrier of R \/ the carrier of S) by A2, A1, ZFMISC_1:96;
A3:
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 A3, RELSET_1:7;
set D = ((IR \/ IS) \/ IQ) \/ IP;
take
RelStr(# ( the carrier of R \/ the carrier of S),(((IR \/ IS) \/ IQ) \/ IP) #)
; ( the carrier of RelStr(# ( the carrier of R \/ the carrier of S),(((IR \/ IS) \/ IQ) \/ IP) #) = the carrier of R \/ the carrier of S & the InternalRel of RelStr(# ( the carrier of R \/ the carrier of S),(((IR \/ IS) \/ IQ) \/ IP) #) = (( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] )
thus
( the carrier of RelStr(# ( the carrier of R \/ the carrier of S),(((IR \/ IS) \/ IQ) \/ IP) #) = the carrier of R \/ the carrier of S & the InternalRel of RelStr(# ( the carrier of R \/ the carrier of S),(((IR \/ IS) \/ IQ) \/ IP) #) = (( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] )
; verum