
theorem lemmultx:
for F being Field
for U,V being VectSp of F
for B being non empty finite Subset of U
for f being Function of B,V
for l1,l2 being Linear_Combination of B
for a being Element of F st l2 = a * l1 holds (f (#) l2) = a * (f (#) l1)
proof
let F be Field, U,V be VectSp of F, B be non empty finite Subset of U;
let f be Function of B,V; let l1,l2 be Linear_Combination of B;
let a be Element of F;
assume AS: l2 = a * l1;
U: dom l1 = the carrier of U & dom l2 = the carrier of U by FUNCT_2:def 1;
now let v be Element of V;
  per cases;
  suppose v in rng f; then
    consider x being object such that
    C1: x in dom f & f.x = v by FUNCT_1:def 3;
    reconsider u = x as Element of B by C1;
    set T1 = Expand(f,l1,v), T2 = Expand(f,l2,v);
KP: len (a*T1) = len T1 by MATRIXR1:16;

 V: len(canFS f"{v}) = card(f"{v}) by FINSEQ_1:93;
    rng(canFS f"{v}) = f"{v} & f"{v} c= B &
      B c= the carrier of U by FUNCT_2:def 3; then
    W: rng(canFS f"{v}) c= dom l1 & rng(canFS f"{v}) c= dom l2 by U; then
    K: len T1 = card(f"{v}) by V,FINSEQ_2:29
             .= len T2 by V,W,FINSEQ_2:29;

    now let i be Nat;
       assume 1 <= i <= len T2;then
       E0:i in dom T2 & i in dom T1 & i in dom (a*T1)
          by FINSEQ_3:25,K,KP;then
       E1: i in dom(canFS f"{v}) by FUNCT_1:11; then
       E2: (canFS f"{v}).i in rng(canFS f"{v}) by FUNCT_1:3;
       rng(canFS f"{v}) = f"{v} & f"{v} c= B &
         B c= the carrier of U by FUNCT_2:def 3;
       then reconsider w = (canFS f"{v}).i as Element of U by E2;
       E3: T1/.i = (l1 * (canFS f"{v})).i by E0,PARTFUN1:def 6
                .= l1.w by E1,FUNCT_1:13;
       E4:T1/.i = T1.i by E0,PARTFUN1:def 6;
       thus T2.i = l2.((canFS f"{v}).i) by E1,FUNCT_1:13
                .= a * T1/.i by E3,AS,VECTSP_6:def 9
                .= (a*T1).i by E4,E0,FVSUM_1:50;
       end; then
 EE: T2 = a * T1 by K,MATRIXR1:16;
    reconsider fl1 = f (#) l1 as Linear_Combination of V;
    thus (f (#) l2).v = Sum T2 by defK
           .= a * (Sum T1) by EE,BINOM:4
           .= a * (fl1.v) by defK
           .= (a * (f (#) l1)).v by VECTSP_6:def 9;
     end;
  suppose C0: not v in rng f;
    hence (f (#) l2).v = a * 0.F by lemadd2b
                      .= a * (f (#) l1).v by C0,lemadd2b
                      .= (a * (f (#) l1)).v by VECTSP_6:def 9;
    end;
  end;
hence thesis by VECTSP_6:def 7;
end;
