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 Th104:
  for s being Tuple of i+j,D
   ex z being Element of i-tuples_on D, t being Element of j-tuples_on D
    st s = z^t
proof
  let s be Tuple of i+j,D;
A1: s is Element of D* by FINSEQ_1:def 11;
  len s = i+j by CARD_1:def 7;
  then s in (i+j)-tuples_on D by A1;
  then s in the set of all z^t where z is Tuple of i,D, t is Tuple of j,D
       by Th103;
  then consider z being Tuple of i,D, t being Tuple of j,D such that
A2:  s = z^t;
   reconsider z as Element of i-tuples_on D by Lm6;
   reconsider t as Element of j-tuples_on D by Lm6;
   s = z^t by A2;
  hence thesis;
end;
