theorem Th56: (P\/Q)|X = P|X \/ (Q|X)
proof
set R1=P|X, R2=Q|X, R=P\/Q, LH=R|X, RH=R1\/R2;
(P null Q)|X c= (P\/Q)|X & (Q null P)|X c= (P\/Q)|X by RELAT_1:76; then
A1: RH c= LH by XBOOLE_1:8;
now
let z be object; assume A2: z in LH; then consider x,y being object such that
A3: z=[x,y] by RELAT_1:def 1;
A4: x in X & [x,y] in (P\/Q) by RELAT_1:def 11, A2, A3;
(x in X & [x,y] in P) or (x in X & [x,y] in Q) by A4, XBOOLE_0:def 3;
then [x,y] in P|X or [x,y] in Q|X by RELAT_1:def 11;
hence z in P|X \/ (Q|X) by XBOOLE_0:def 3, A3;
end; then LH c= RH; hence thesis by A1;
end;
