reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;
reserve s for Element of D*;

theorem Th103:
  (i+j)-tuples_on D = the set of all z^t where z is Tuple of i,D,
  t is Tuple of j,D
proof
  set T = the set of all z^t where z is Tuple of i,D, t is Tuple of j,D;
  now
    let x be object;
    thus x in (i+j)-tuples_on D implies x in T
    proof
      assume x in (i+j)-tuples_on D;
      then consider s such that
A1:   x = s and
A2:   len s = i + j;
      consider z,t being FinSequence of D such that
A3:   len z = i & len t = j and
A4:   s = z^t by A2,Th21;
      z is Tuple of i,D & t is Tuple of j,D by A3,CARD_1:def 7;
      hence thesis by A1,A4;
    end;
    assume x in T;
    then consider z being Tuple of i,D, t being Tuple of j,D such that
A5: x = z^t;
    len z = i & len t = j by CARD_1:def 7;
    then
A6: len(z^t) = i+j by FINSEQ_1:22;
    z^t is Element of D* by FINSEQ_1:def 11;
    hence x in (i+j)-tuples_on D by A5,A6;
  end;
  hence thesis by TARSKI:2;
end;
