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;

theorem Th1: m-tuples_on A /\ (B*) = m-tuples_on A /\ (m-tuples_on B)
proof
reconsider mm=m as Element of NAT by ORDINAL1:def 12;
set L=m-tuples_on A /\ (B*), R=m-tuples_on A /\ m-tuples_on B;
m-tuples_on A /\ L c= R by XBOOLE_1:26, Lm8; then
A1: (m-tuples_on A /\ m-tuples_on A) /\ (B*) c= R by XBOOLE_1:16;
now
let x be object; assume A2: x in m-tuples_on B; then
reconsider xx=x as m-element FinSequence by FINSEQ_2:141;
xx in mm-tuples_on B by A2;then len xx=mm & rng xx c= B by FINSEQ_2:132;then
x is FinSequence of B by FINSEQ_1:def 4;hence x in B* by FINSEQ_1:def 11;
end;
then R c= L by XBOOLE_1:26, TARSKI:def 3;
hence thesis by  A1;
end;
