
theorem ThMappingFrobProd2:
  for G1, G2 being Group
  for phi being Homomorphism of G1, G2
  for F1 being FinSequence of the carrier of G1
  for ks being FinSequence of INT
  ex F2 being FinSequence of the carrier of G2
  st len F1 = len F2
   & F2 = phi * F1
   & Product (F2 |^ ks) = phi.(Product (F1 |^ ks))
proof
  let G1, G2 be Group;
  let phi be Homomorphism of G1, G2;
  let F1 be FinSequence of the carrier of G1;
  let ks be FinSequence of INT;
  consider F2 being FinSequence of the carrier of G2 such that
  A2: len F1 = len F2 & F2 = phi * F1 & Product F2 = phi.(Product F1)
  by ThMappingFrobProd;
  take F2;
  thus len F1 = len F2 & F2 = phi * F1 by A2;
  A3: len (F1 |^ ks) = len (F1) by GROUP_4:def 3
                    .= len (F2 |^ ks) by A2, GROUP_4:def 3;
  A4: len (phi * (F1 |^ ks)) = len (F1 |^ ks) by FINSEQ_2:33;
  for k being Nat st k in dom (F2 |^ ks)
  holds (phi * (F1 |^ ks)).k = (F2 |^ ks).k
  proof
    let k be Nat;
    assume k in dom (F2 |^ ks);
    then k in Seg (len (F1 |^ ks)) by A3, FINSEQ_1:def 3;
    then B1: k in dom (F1 |^ ks) by FINSEQ_1:def 3;
    then k in Seg (len (F1 |^ ks)) by FINSEQ_1:def 3;
    then k in Seg (len (F1)) by GROUP_4:def 3;
    then B2: k in dom F1 by FINSEQ_1:def 3;
    then k in Seg (len (F2)) by A2, FINSEQ_1:def 3;
    then B3: k in dom F2 by FINSEQ_1:def 3;
    B4: F1 /. k in G1 & dom phi = the carrier of G1 by FUNCT_2:def 1;
    set g = (F1 /. k);
    set n = (@ (ks /. k));
    (phi * (F1 |^ ks)).k = phi.((F1 |^ ks).k) by B1, FUNCT_1:13
                        .= phi.(g |^ n) by B2, GROUP_4:def 3
                        .= (phi /. g) |^ n by GROUP_6:37
                        .= ((phi * F1)/.k) |^ n by B2, B4, PARTFUN2:4
                        .= (F2 |^ ks).k by A2, B3, GROUP_4:def 3;
    hence (phi * (F1 |^ ks)).k = (F2 |^ ks).k;
  end;
  hence thesis by A3, A4, ThMappingFrobProdProperty, FINSEQ_2:9;
end;
