reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem Th28:
  f (#) (F ^ G) = (f (#) F) ^ (f (#) G)
proof
  set H = (f (#) F) ^ (f (#) G);
  set I = F ^ G;
A1: len F = len(f (#) F) by Def6;
A2: len H = len(f (#) F) + len(f (#) G) by FINSEQ_1:22
    .= len F + len(f (#) G) by Def6
    .= len F + len G by Def6
    .= len I by FINSEQ_1:22;
A3: len G = len(f (#) G) by Def6;
  now
    let k be Nat;
    assume
A4: k in dom H;
    now
      per cases by A4,FINSEQ_1:25;
      suppose
A5:     k in dom(f (#) F);
        then
A6:     k in dom F by A1,FINSEQ_3:29;
        then
A7:     k in dom(F ^ G) by FINSEQ_3:22;
A8:     F/.k = F.k by A6,PARTFUN1:def 6
          .= (F ^ G).k by A6,FINSEQ_1:def 7
          .= (F ^ G)/.k by A7,PARTFUN1:def 6;
        thus H.k = (f (#) F).k by A5,FINSEQ_1:def 7
          .= (I/.k) * f.(I/.k) by A5,A8,Def6;
      end;
      suppose
A9:     ex n be Nat st n in dom(f (#) G) & k = len(f (#) F) + n;
A10:    k in dom I by A2,A4,FINSEQ_3:29;
        consider n be Nat such that
A11:    n in dom(f (#) G) and
A12:    k = len(f (#) F) + n by A9;
A13:    n in dom G by A3,A11,FINSEQ_3:29;
        then
A14:    G/.n = G.n by PARTFUN1:def 6
          .= (F ^ G).k by A1,A12,A13,FINSEQ_1:def 7
          .= (F ^ G)/.k by A10,PARTFUN1:def 6;
        thus H.k = (f (#) G).n by A11,A12,FINSEQ_1:def 7
          .= (I/.k) * f.(I/.k) by A11,A14,Def6;
      end;
    end;
    hence H.k = (I/.k) * f.(I/.k);
  end;
  hence thesis by A2,Def6;
end;
