reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;

theorem Th24:
  l = l!Carrier l
proof
  set f = l|(Carrier l);
  set g = (V \ Carrier l) --> 0.F;
  set m = f +* g;
A2: dom l = [#]V by FUNCT_2:92;
  then dom f = Carrier l by RELAT_1:62;
  then
A3: (dom f) \/ (dom g) = [#]V by XBOOLE_1:45;
A4: for x being object st x in dom l holds l.x = m.x
  proof
    let x be object;
    assume x in dom l;
    then reconsider x as Element of V;
    per cases;
    suppose
A5:   x in Carrier l;
      then not x in dom g by XBOOLE_0:def 5;
      then m.x = f.x by A3,FUNCT_4:def 1;
      hence thesis by A5,FUNCT_1:49;
    end;
    suppose
A6:   not x in Carrier l;
      then x in V \ (Carrier l) by XBOOLE_0:def 5;
      then m.x = g.x & g.x = 0.F by A3,FUNCOP_1:7,FUNCT_4:def 1;
      hence thesis by A6;
    end;
  end;
  dom l = dom m by A2,A3,FUNCT_4:def 1;
  hence thesis by A4;
end;
