 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve s for FinSequence,
   V1,V2,V3 for finite-rank free Z_Module,
   f,f1,f2 for Function of V1,V2,
   g for Function of V2,V3,
   b1 for OrdBasis of V1,
   b2 for OrdBasis of V2,
   b3 for OrdBasis of V3,
   v1,v2 for Vector of V2,
   v,w for Element of V1;
 reserve p2,F for FinSequence of V1,
   p1,d for FinSequence of INT.Ring,
   KL for Linear_Combination of V1;

theorem Th19:
  for F, F1 being FinSequence of V1, KL being Linear_Combination of V1,
  p being Permutation of dom F st F1 = F * p holds KL (#) F1 = (KL (#) F) * p
  proof
    let F, F1 be FinSequence of V1;
    let KL be Linear_Combination of V1;
    let p be Permutation of dom F such that
    A1: F1 = F * p;
    dom F = Seg len F by FINSEQ_1:def 3;
    then dom F = Seg len (KL (#) F) by VECTSP_6:def 5;
    then
    A2: dom F = dom (KL (#) F) by FINSEQ_1:def 3;
    then reconsider F2 = (KL (#) F) * p as FinSequence of V1 by FINSEQ_2:47;
    A31: len (KL (#) F1) = len F1 by VECTSP_6:def 5
    .= len F by A1,FINSEQ_2:44
    .= len (KL (#) F) by VECTSP_6:def 5
    .= len F2 by A2,FINSEQ_2:44;
    then
    A3: dom (KL (#) F1) = dom ((KL (#) F) * p) by FINSEQ_3:29;
    len (KL (#) F1) = len F1 by VECTSP_6:def 5;
    then
    A4: dom (KL (#) F1) = dom F1 by FINSEQ_3:29;
    now
      let k be Nat;
      reconsider k0 = k as Element of NAT by ORDINAL1:def 12;
      assume
      A5: k in dom (KL (#) F1);
      then k in dom p by A3,FUNCT_1:11;
      then A6: p.k in rng p by FUNCT_1:def 3;
      then p.k in dom F;
      then reconsider k1 = p.k0 as Element of NAT;
      F1/.k = F1.k by A4,A5,PARTFUN1:def 6
      .= F.(p.k) by A1,A4,A5,FUNCT_1:12
      .= F/.(p.k) by A6,PARTFUN1:def 6;
      hence (KL (#) F1).k = KL.(F/.k1) * (F/.k1) by A5,VECTSP_6:def 5
      .= (KL (#) F).k1 by A2,A6,VECTSP_6:def 5
      .= F2.k by A3,A5,FUNCT_1:12;
    end;
    hence thesis by A31,FINSEQ_3:29;
  end;
