reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;

theorem Th40:
  for n being non zero Nat, X being non empty set,
  S being non empty Subset-Family of X st S <> {{}} holds
  Product(n,S) c= bool Funcs(Seg n,X)
  proof
    let n be non zero Nat,X be non empty set,S be non empty Subset-Family of X;
    assume
A1: S <> {{}};
A2: Product(n,S) c= bool Funcs(Seg n,union S) by Th39;
    union S c= union bool X by ZFMISC_1:77; then
A3: union S is non empty Subset of X by A1,ZFMISC_1:81,SCMYCIEL:18;
    n-tuples_on X = Funcs(Seg n,X) &
      n-tuples_on union S = Funcs(Seg n,union S) by FINSEQ_2:93;
    then bool Funcs(Seg n,union S) c= bool Funcs(Seg n,X)
      by A3,Th5,ZFMISC_1:67;
    hence thesis by A2;
  end;
