
theorem LSum1a:
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
for b being Element of K holds down(l,b) = down(l1,b) + down(l2,b)
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;
let b be Element of K;
E is Subring of K by FIELD_4:def 1; then
X1: the carrier of E c= the carrier of K by C0SP1:def 3;
now let o be object;
  assume A0: o in the carrier of VecSp(E,F); then
  A2: o in the carrier of E by FIELD_4:def 6; then
  reconsider a = o as Element of K by X1;
  reconsider aE = o as Element of E by A0,FIELD_4:def 6;
  reconsider aV = o as Element of VecSp(E,F) by A0;
  reconsider abV = a * b as Element of VecSp(K,F) by FIELD_4:def 6;
  per cases;
  suppose A1: a in BE & b in BK;
    then down(l,b).a
           = l.(a*b) by down1
          .= l1.abV + l2.abV by AS,VECTSP_6:22
          .= (the addF of F).[down(l1,b).a,l2.abV] by A1,down1
          .= down(l1,b).aV + down(l2,b).aV by A1,down1
          .= (down(l1,b) + down(l2,b)).aV by VECTSP_6:22;
    hence down(l,b).o = (down(l1,b) + down(l2,b)).o;
    end;
  suppose A1: not a in BE or not b in BK;
    then A3: down(l1,b).a = 0.F & down(l2,b).a = 0.F by A2,down1;
    down(l,b).a = down(l1,b).aV + down(l2,b).aV by A3,A1,A2,down1;
    hence down(l,b).o = (down(l1,b) + down(l2,b)).o by VECTSP_6:22;
    end;
  end;
hence thesis;
end;
