
theorem LSum1:
for F being Field,
    E being FieldExtension of F,
    K being F-extending FieldExtension of E
for BE being non empty finite linearly-independent Subset of VecSp(E,F),
    BK being non empty finite linearly-independent Subset of VecSp(K,E)
for l,l1,l2 being Linear_Combination of Base(BE,BK) st l = l1 + l2
holds down l = down l1 + down l2
proof
let F be Field, E be FieldExtension of F,
    K be F-extending FieldExtension of E;
let BE be non empty finite linearly-independent Subset of VecSp(E,F),
    BK be non empty finite linearly-independent Subset of VecSp(K,E);
let l,l1,l2 be Linear_Combination of Base(BE,BK);
assume AS: l = l1 + l2;
now let o be object;
  assume A0: o in the carrier of VecSp(K,E); then
  reconsider b = o as Element of K by FIELD_4:def 6;
  reconsider bV = o as Element of VecSp(K,E) by A0;
  H1: Carrier(down l) c= BK & Carrier(down l1) c= BK &
      Carrier(down l2) c= BK by VECTSP_6:def 4;
  per cases;
  suppose A1: b in BK;
    then A2: (down l1).b = Sum down(l1,b) by down2;
    (down l).b = Sum down(l,b) by A1,down2
              .= Sum(down(l1,b) + down(l2,b)) by AS,LSum1a
              .= Sum down(l1,b) + Sum down(l2,b) by VECTSP_6:44
              .= (the addF of E).[Sum down(l1,b),Sum down(l2,b)]
                 by FIELD_4:def 6
              .= (down l1).bV + (down l2).bV by A1,A2,down2;
    hence (down l).o = (down l1 + down l2).o by VECTSP_6:22;
    end;
  suppose A1: not b in BK;
    then (down l).bV = 0.E by H1,VECTSP_6:2
                    .= (down l1).bV + 0.E by A1,H1,VECTSP_6:2
                    .= (down l1).bV + (down l2).bV by A1,H1,VECTSP_6:2;
    hence (down l).o = (down l1 + down l2).o by VECTSP_6:22;
    end;
  end;
hence thesis;
end;
