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 x in m-tuples_on A implies x is FinSequence of A
proof
assume A1: x in m-tuples_on A; then
reconsider p=x as m-element FinSequence by FINSEQ_2:141;
p in Funcs(Seg m,A) by A1, Lm7; then consider f being Function
such that A2: p=f & dom f = Seg m & rng f c= A by FUNCT_2:def 2;
thus x is FinSequence of A by A2, FINSEQ_1:def 4;
end;
