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 (A/\B)* = A* /\ (B*)
proof
set X=A/\B; reconsider XA=A/\B as Subset of A; reconsider XB=A/\B as
Subset of B; XA* c= A* & XB* c= B*; then A1: X* c= A* /\ (B*) by XBOOLE_1:19;
now
let x be object; assume
A2: x in A*/\(B*); reconsider pa=x as A-valued FinSequence by A2;
set m=len pa, mA=m-tuples_on A, mB=m-tuples_on B, mX=m-tuples_on X;
mX\(X*)={}; then A3: mX c= X* by XBOOLE_1:37;
reconsider pb=x as B-valued FinSequence by A2;
pa is m-element & pb is m-element by CARD_1:def 7; then
pa in mA & pb in mB by Th16; then pa in mA/\mB by XBOOLE_0:def 4; then
pa in mX by Th3; hence x in X* by A3;
end; then A*/\(B*) c= X*; hence thesis by A1;
end;
