
theorem lemadd2:
for F being Field
for U,V being VectSp of F
for B being non empty finite Subset of U
for b being Element of B
for f being Function of B,V
for l being Linear_Combination of B st Carrier l = {b}
holds Carrier(f (#) l) = {f.b} & Sum(f (#) l) = l.b * f.b
proof
let F be Field, U,V be VectSp of F;
let B be non empty finite Subset of U, b be Element of B;
let f be Function of B,V, l being Linear_Combination of B;
assume A: Carrier l = {b};
H: dom f = B by FUNCT_2:def 1; then
reconsider v = f.b as Element of (rng f) by FUNCT_1:3;
F: b in Carrier l by A,TARSKI:def 1;
set T = Expand(f,l,v);
f.b in {v} by TARSKI:def 1; then
b in f"{v} by H,FUNCT_1:def 7; then
b in rng canFS(f"{v}) by FUNCT_2:def 3; then
consider i being object such that
E: i in dom canFS(f"{v}) & (canFS(f"{v})).i = b by FUNCT_1:def 3;
reconsider i as Element of NAT by E;
K: dom l = the carrier of U by FUNCT_2:def 1; then
L: i in dom T by E,FUNCT_1:11;
G: l.b = (l * (canFS(f"{v}))).i by E,FUNCT_1:13
      .= T/.i by L,PARTFUN1:def 6;
I: now let j be Element of NAT;
   assume I1: j in dom T & j <> i; then
   I2: j in dom canFS(f"{v}) & (canFS(f"{v})).j in dom l by FUNCT_1:11; then
   (canFS(f"{v})).j <> b by E,I1,FUNCT_1:def 4; then
   not (canFS(f"{v})).j in Carrier l by A,TARSKI:def 1;
   hence 0.F = l.((canFS(f"{v})).j) by I2
            .= (l * (canFS(f"{v}))).j by I2,FUNCT_1:13
            .= T/.j by I1,PARTFUN1:def 6;
   end;
C: (f (#) l).v = Sum T by defK .= l.b by G,K,I,E,FUNCT_1:11,POLYNOM2:3;
B: now let o be object;
   assume o in {f.b}; then
   B0: o = v by TARSKI:def 1;
   (f (#) l).v <> 0.F by C,F,VECTSP_6:2;
   hence o in Carrier(f (#) l) by B0;
   end;
D: now let o be object;
   assume o in Carrier(f (#) l); then
   consider v being Element of V such that
   B0: o = v & (f (#) l).v <> 0.F;
   B1: f .: {b} = Im(f,b) .= {f.b} by H,FUNCT_1:59;
   B2: Carrier(f (#) l) c= f .: {b} by A,lemadd2a;
   now assume v <> f.b;
    then not v in Carrier(f (#) l) by B2,B1,TARSKI:def 1;
     hence contradiction by B0;
     end;
   hence o in {f.b} by B0,TARSKI:def 1;
   end;
hence Carrier(f (#) l) = {f.b} by B,TARSKI:2;
thus Sum(f (#) l) = l.b * f.b by D,C,B,TARSKI:2,VECTSP_6:20;
end;
