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 Th47: for A,B being functional set holds
SymbolsOf (A\/B) = SymbolsOf A \/ (SymbolsOf B)
proof
let A, B be functional set;
set AF={rng x where x is Element of A: x in A}, BF=
{rng x where x is Element of B: x in B}, F=
{rng x where x is Element of A\/B: x in A\/B};
A null B c= A\/B & B null A c= A\/B; then reconsider
AFF=AF, BFF=BF as Subset of F by Lm51;
A1: AFF \/ BFF c= F;
now
let y be object; assume y in F\BF; then
A2: y in F & not y in BF by XBOOLE_0:def 5; then
consider x being Element of A\/B such that
A3: y=rng x & x in A\/B;
not x in B by A3, A2;
then
A4: x in A null {{}} by A3, XBOOLE_0:def 3; then reconsider xx=x as
Element of A\/{{}};
thus y in AF by A4, A3;
end; then
F\BF \/ BF c= AF \/ BF by XBOOLE_1:9, TARSKI:def 3;
then F null BFF c= AF \/ BF by XBOOLE_1:39; then
A5: AF \/ BF = F by A1;
thus thesis by A5, ZFMISC_1:78;
end;
