
theorem fact2:
for R being well-unital associative non empty doubleLoopStr,
    a,b being Element of R,
    F,G being non empty FinSequence of R
st F is_a_factorization_of a & G is_a_factorization_of b
holds F^G is_a_factorization_of a * b
proof
let R be well-unital associative non empty doubleLoopStr,
    a,b be Element of R,
    F,G be non empty FinSequence of R;
assume AS: F is_a_factorization_of a & G is_a_factorization_of b;
reconsider FG = F^G as non empty FinSequence of R;
rng F c= the carrier of R; then
reconsider f = F as Function of dom F,R by FUNCT_2:2;
rng G c= the carrier of R; then
reconsider g = G as Function of dom G,R by FUNCT_2:2;
rng FG c= the carrier of R; then
reconsider fg = FG as Function of dom FG,R by FUNCT_2:2;
now let i be Element of dom(F^G);
  now per cases by FINSEQ_1:25;
  suppose B: i in dom F;
     fg.i = f.i by B,FINSEQ_1:def 7;
     hence (F^G).i is irreducible by B,AS;
     end;
  suppose ex n being Nat st n in dom G & i = len F + n;
     then consider n being Nat such that B: n in dom G & i = len F + n;
     fg.i = g.n by B,FINSEQ_1:def 7;
     hence (F^G).i is irreducible by B,AS;
     end;
  end;
  hence (F^G).i is irreducible;
  end;
hence thesis by AS,GROUP_4:5;
end;
