
theorem lemadd2a:
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 l being Linear_Combination of B holds Carrier(f (#) l) c= f .: (Carrier l)
proof
let F be Field, U,V be VectSp of F;
let B be non empty finite Subset of U;
let f be Function of B,V, l be Linear_Combination of B;
now let o be object;
  assume o in Carrier(f (#) l); then
  consider v being Element of V such that
  B1: o = v & (f (#) l).v <> 0.F;
  set T = Expand(f,l,v);
  now assume A3: f"{v} misses Carrier l;
    set L = (len T |-> 0.F);
    now let i be Nat;
      assume 1 <= i <= len T; then
      A6: i in dom T & i in Seg len T by FINSEQ_3:25; then
      i in dom(canFS f"{v}) & (canFS f"{v}).i in dom l by FUNCT_1:11; then
      A4: (canFS f"{v}).i in rng(canFS f"{v}) by FUNCT_1:3;
      f"{v} c= B; then
      A5: (canFS f"{v}).i in B by A4;
      not (canFS f"{v}).i in Carrier l by A4,A3,XBOOLE_0:def 4;
      then 0.F = l.((canFS f"{v}).i) by A5
               .= T.i by A6,FUNCT_1:12;
      hence T.i = L.i by A6, FINSEQ_2:57;
      end; then
    T=L by CARD_1:def 7;then
    Sum T = 0.F by MATRIX_3:11;
    hence contradiction by B1,defK;
    end; then
  consider y being object such that
  A8: y in f"{v} and
  A9: y in Carrier l by XBOOLE_0:3;
  A10: y in dom f by A8,FUNCT_1:def 7;
  A11: f.y in {v} by A8,FUNCT_1:def 7;
  reconsider y as Element of B by A8;
  f.y = v by A11,TARSKI:def 1;
  hence o in f .: (Carrier l) by B1,A9,A10,FUNCT_1:def 6;
  end;
hence thesis;
end;
