reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;
reserve f for BinOp of D;
reserve a,a1,a2,b,b1,b2,A,B,C,X,Y,Z,x,x1,x2,y,y1,y2,z for set,
U,U1,U2,U3 for non empty set, u,u1,u2 for Element of U,
P,Q,R for Relation, f,f1,f2,g,g1,g2 for Function,
k,m,n for Nat, kk,mm,nn for Element of NAT, m1, n1 for non zero Nat,
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;

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;
