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;
