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;
reserve n for Nat,
        S for Subset-Family of REAL;

theorem Th42:
  for S being non empty Subset-Family of REAL, x being Subset of REAL n holds
    x is Element of Product(n,S)
      iff
    ex s being Tuple of n,S st
      for t being Element of REAL n holds
         (for i being Nat st i in Seg n holds t.i in s.i) iff t in x
  proof
    let S be non empty Subset-Family of REAL, x being Subset of REAL n;
A1: x is Subset of Funcs(Seg n,REAL) by FINSEQ_2:93;
    thus x is Element of Product(n,S) implies (ex s be Tuple of n,S st
      (for t be Element of REAL n holds (for i be Nat st i in Seg n holds
       t.i in s.i) iff (t in x)))
    proof
      assume x is Element of Product(n,S);
      then consider s be Tuple of n,S such that
A2:   (for t be Element of Funcs(Seg n,REAL) holds (for i be Nat st
        i in Seg n holds t.i in s.i) iff (t in x)) by Lm1;
      take s;
      for t be Element of REAL n holds (for i be Nat st i in Seg n holds
        t.i in s.i) iff (t in x)
      proof
        let t be Element of REAL n;
        t is Element of Funcs(Seg n,REAL) by FINSEQ_2:93;
        hence thesis by A2;
      end;
      hence thesis;
    end;
    thus (ex s be Tuple of n,S st (for t be Element of REAL n holds
      (for i be Nat st i in Seg n holds t.i in s.i) iff (t in x))) implies
      x is Element of Product(n,S)
    proof
      given s be Tuple of n,S such that
A3:   (for t be Element of REAL n holds (for i be Nat st i in Seg n holds
        t.i in s.i) iff t in x);
      ex s be Tuple of n,S st for t be Element of Funcs(Seg n,REAL) holds
       (for i be Nat st i in Seg n holds t.i in s.i) iff (t in x)
      proof
        take s;
        let t be Element of Funcs(Seg n,REAL);
        t is Element of REAL n by FINSEQ_2:93;
        hence (for i be Nat st i in Seg n holds t.i in s.i) iff (t in x) by A3;
      end;
      hence thesis by A1,Lm2;
    end;
  end;
