reserve
  I for non empty set,
  F for associative Group-like multMagma-Family of I,
  i, j for Element of I;
reserve n for non zero Nat;

theorem
  for G,F being associative commutative Group-like multMagma-Family of Seg n
  st
  for k be Element of Seg n holds F.k = ProjGroup(G,k)
  holds
  ex f being Homomorphism of product F,product G
  st f is bijective &
  for x be Element of product F
  ex s be FinSequence of product G st len s= n &
  (for k be Element of Seg n holds s.k in F.k) & s=x & f.x = Product s
  proof
    let G,F be associative commutative Group-like
         multMagma-Family of Seg n;
    assume A1: for k be Element of Seg n
    holds F.k = ProjGroup(G,k);
A2: for i be Element of Seg n holds F.i is Subgroup of product G
    proof
      let i be Element of Seg n;
      F.i = ProjGroup(G,i) by A1;
      hence thesis;
    end;
A3: for x be Element of product G
    ex s be FinSequence of product G st len s = n
    & (for k be Element of Seg n holds s.k in F.k) & x=Product s
    proof
      let x be Element of product G;
      consider s be FinSequence of product G such that
      A4: len s = n
      & (for k be Element of Seg n holds s.k in ProjGroup(G,k)) &
      x=Product s by Th11;
      take s;
      for k be Element of Seg n holds s.k in F.k
      proof
        let k be Element of Seg n;
        s.k in ProjGroup(G,k) by A4;
        hence thesis by A1;
      end;
      hence thesis by A4;
    end;
    for s,t be FinSequence of product G st
    len s = n & (for k be Element of Seg n holds s.k in F.k) & len t = n
    & (for k be Element of Seg n holds t.k in F.k)
    & Product s=Product t holds s=t
    proof
      let s,t be FinSequence of product G;
      assume A5:
      len s = n & (for k be Element of Seg n holds s.k in F.k) &
      len t = n & (for k be Element of Seg n holds t.k in F.k) &
      Product s=Product t;
      for k be Element of Seg n holds
      t.k in ProjGroup(G,k) & s.k in ProjGroup(G,k)
      proof
        let k be Element of Seg n;
        s.k in F.k & t.k in F.k by A5;
        hence thesis by A1;
      end;
      hence thesis by A5,Th10;
    end;
    hence thesis by A2,A3,Th12;
  end;
