reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;
reserve n for Nat;
reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem Th129:
 for D being set holds product (i |-> D) = i-tuples_on D
proof let D be set;
  thus product (i |-> D) = product (Seg i --> D)
    .= Funcs(Seg i,D) by CARD_3:11
    .= i-tuples_on D by FINSEQ_2:93;
end;
