 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 Th17:
  for a being FinSequence of INT.Ring, p being FinSequence of V1
  st len p = len a holds f is additive homogeneous
  implies f*lmlt(a,p) = lmlt(a,f*p)
  proof
    let a be FinSequence of INT.Ring, p be FinSequence of V1;
    assume len p = len a;
    then
    A1: dom p = dom a by FINSEQ_3:29;
    dom f = the carrier of V1 by FUNCT_2:def 1;
    then rng p c= dom f;
    then
    A2: dom p = dom (f*p) by RELAT_1:27;
    assume
    A3: f is additive homogeneous;
    A4:
    now
      set P = f*p;
      let k be Nat;
      assume
      A5: k in dom (f*lmlt(a,p));
      A6: dom (f*lmlt(a,p)) c= dom lmlt(a,p) by RELAT_1:25;
      then k in dom lmlt(a,p) by A5;
      then
      A7: k in dom p by A1,Th12;
      then
      A8: p/.k = p.k by PARTFUN1:def 6;
      A9: k in dom lmlt(a,f*p) by A1,A2,A7,Th12;
      A10: a/.k = a.k by A1,A7,PARTFUN1:def 6;
      A11: P/.k = (f*p).k by A2,A7,PARTFUN1:def 6;
      thus (f*lmlt(a,p)).k = f.(lmlt(a,p).k) by A5,FUNCT_1:12
      .= f.((a/.k)*(p/.k)) by A10,A8,A5,A6,FUNCOP_1:22
      .= (a/.k)*(f.(p/.k)) by A3
      .= (a/.k)*(P/.k) by A7,A8,A11,FUNCT_1:13
      .= lmlt(a,f*p).k by A9,A10,A11,FUNCOP_1:22;
    end;
    dom lmlt(a,p) = dom p by A1,Th12
    .= dom lmlt(a,f*p) by A1,A2,Th12;
    then len lmlt(a,p) = len lmlt(a,f*p) by FINSEQ_3:29;
    then len (f*lmlt(a,p)) = len lmlt(a,f*p) by FINSEQ_2:33;
    hence thesis by A4,FINSEQ_2:9;
  end;
