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;

theorem Th23: for P,Q being Relation holds (P\/Q)"Y = P"Y \/ (Q"Y)
proof
let P,Q be Relation; set R=P\/Q, LH=R"Y, RH=P"Y \/ (Q"Y);
reconsider PP=P null Q, QQ=Q null P as Subset of R;
now
let x be object; assume x in LH; then consider y being object such that
A1: [x,y] in R & y in Y by RELAT_1:def 14;set z=[x,y];
(z in P & y in Y) or(z in Q & y in Y) by XBOOLE_0:def 3, A1; then
x in P"Y or x in Q"Y by RELAT_1:def 14;hence x in RH by XBOOLE_0:def 3;
end; then
A2: LH c= RH; reconsider PX=PP"Y, QX=QQ"Y as Subset of LH
by RELAT_1:144; PX \/ QX c= LH; hence thesis by A2;
end;
