reserve V,W for Z_Module;
reserve T for linear-transformation of V,W;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;
reserve V,W for Z_Module;
reserve l for Linear_Combination of V;
reserve T for linear-transformation of V,W;

theorem Th37:
  for R being Ring
  for V,W being LeftMod of R
  for l being Linear_Combination of V,
      T being linear-transformation of V,W
  for v being Element of V st T | (Carrier l) is one-to-one &
  v in Carrier l holds (T@*l).(T.v) = l.v
  proof
    let R be Ring;
    let V,W be LeftMod of R;
    let l be Linear_Combination of V,
        T be linear-transformation of V,W;
    let v be Element of V such that
    A1: T | (Carrier l) is one-to-one and
    A2: v in Carrier l;
    v in the carrier of V; then
    A3: v in dom l by FUNCT_2:def 1;
    A4: dom T = the carrier of V by FUNCT_2:def 1;
    for x be object holds
    x in T"{T.v} /\ Carrier(l) iff x in {v}
    proof
      let x be object;
      hereby assume x in T"{T.v} /\ Carrier(l); then
        A5: x in T"{T.v} & x in Carrier(l) by XBOOLE_0:def 4; then
        A6: x in the carrier of V & T.x in {T.v} by FUNCT_2:38;
        A7: T | (Carrier l).x = T.x by A5,FUNCT_1:49
        .= T.v by A6,TARSKI:def 1
        .= T | (Carrier l).v by A2,FUNCT_1:49;
        A8: x in dom(T | (Carrier l)) by A4,A5,RELAT_1:57;
        v in dom(T | (Carrier l)) by A2,A4,RELAT_1:57;
        then x = v by A1,A7,A8,FUNCT_1:def 4;
        hence x in {v} by TARSKI:def 1;
      end;
      assume x in {v}; then
      A9: x = v by TARSKI:def 1;
      x in the carrier of V & T.x in {T.v} by A9,TARSKI:def 1;
      then x in T"{T.v} by FUNCT_2:38;
      hence thesis by A2,A9,XBOOLE_0:def 4;
    end;
    then T"{T.v} /\ Carrier(l) = {v} by TARSKI:2;
    then canFS((T"{T.v}) /\ (Carrier l)) = <*v*> by FINSEQ_1:94;
    then lCFST(l,T,T.v) = <*l.v*> by A3,FINSEQ_2:34;
    then Sum(lCFST(l,T,T.v)) = l.v by RLVECT_1:44;
    hence thesis by LDef5;
  end;
