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
  for R being Ring
  for V,W be LeftMod of R, X be Subset of V,
      T be linear-transformation of V,W,
      X being Subset of V,
      l be Linear_Combination of T .: X st
  T|X is one-to-one holds T@*(T#l) = l
  proof
    let R be Ring;
    let V,W be LeftMod of R, X be Subset of V,
        T be linear-transformation of V,W;
    let X be Subset of V,
        l be Linear_Combination of T .: X such that
    A1: T|X is one-to-one;
    let w be Element of W;
    set m = T@*(T#l);
    reconsider SZ0 = {0.R} as finite Subset of R;
    per cases;
    suppose
      A2: w in Carrier l;
      Carrier l c= T .: X by VECTSP_6:def 4;
      then consider v being object such that
      A3: v in dom T and
      A4: v in X and
      A5: w = T.v by A2,FUNCT_1:def 6;
      reconsider v as Element of V by A3;
      w in the carrier of W; then
      A6: w in dom l by FUNCT_2:def 1;
      A7: v in the carrier of V;
      for x be object holds
      x in T"{w} /\ Carrier((T#l)) iff x in {v} /\ Carrier((T#l))
      proof
        let x be object;
        hereby
          assume
          A8: x in T"{w} /\ Carrier((T#l)); then
          A9: x in T"{w} & x in Carrier((T#l)) by XBOOLE_0:def 4; then
          A10: x in X by TARSKI:def 3,VECTSP_6:def 4;
          x in the carrier of V by A8; then
          A11: x in dom T by FUNCT_2:def 1;
          A12: x in the carrier of V & T.x in {w} by A9,FUNCT_2:38;
          A13:(T|X).x = T.x by A10,FUNCT_1:49
          .= T.v by A5,A12,TARSKI:def 1
          .= (T|X).v by A4,FUNCT_1:49;
          A14: x in dom (T|X) by A10,A11,RELAT_1:57;
          v in dom (T|X) by A3,A4,RELAT_1:57;
          then x = v by A1,A13,A14,FUNCT_1:def 4;
          then x in {v} by TARSKI:def 1;
          hence x in {v} /\ Carrier((T#l)) by A9,XBOOLE_0:def 4;
        end;
        assume x in {v} /\ Carrier((T#l)); then
        A15: x in {v} & x in Carrier((T#l)) by XBOOLE_0:def 4; then
        A16: x = v by TARSKI:def 1;
        x in the carrier of V & T.x in {w} by A5,A16,TARSKI:def 1;
        then x in T"{w} by FUNCT_2:38;
        hence x in T"{w} /\ Carrier((T#l)) by A15,XBOOLE_0:def 4;
      end; then
      A17: T"{w} /\ Carrier((T#l)) = {v} /\ Carrier((T#l)) by TARSKI:2;
      per cases;
      suppose
        A18: not v in Carrier((T#l));
        {v} /\ Carrier((T#l)) = {}
        proof
          assume {v} /\ Carrier((T#l)) <> {};
          then consider x be object such that
          A19: x in {v} /\ Carrier((T#l)) by XBOOLE_0:def 1;
          A20: x in {v} by A19,XBOOLE_0:def 4;
          x in Carrier((T#l)) by A19,XBOOLE_0:def 4;
          hence contradiction by A18,A20,TARSKI:def 1;
        end; then
b1:     lCFST(T#l,T,w) = <*>the carrier of R by A17;
        (T@*(T#l)).w = Sum(lCFST(T#l,T,w)) by LDef5; then
   A21: (T@*(T#l)).w = 0.R by RLVECT_1:43,b1;
        A22:(T#l).v = 0.R by A18;
        A23: not v in dom((X`) --> 0.R) by A4,XBOOLE_0:def 5;
        (T#l) = (l*T) +* ((X`) --> 0.R) by A1,Def6;
        then (T#l).v = (l*T).v by A23,FUNCT_4:11;
        hence (T@*(T#l)).w = l.w by A3,A5,A21,A22,FUNCT_1:13;
      end;
      suppose
        v in Carrier((T#l));
        then T"{w} /\ Carrier(T#l) = {v} by A17,XBOOLE_1:28,ZFMISC_1:31;
        then
        A24: canFS((T"{w}) /\ (Carrier (T#l))) = <*v*> by FINSEQ_1:94;
        A25: not v in dom((X`) --> 0.R) by A4,XBOOLE_0:def 5;
        A26: (T#l) = (l*T) +* ((X`) --> 0.R) by A1,Def6;
        A27: v in dom(T#l) by A7,FUNCT_2:def 1;
        A28: v in dom(l*T) by A7,FUNCT_2:def 1;
        (T#l) * (<*v*>) = <* (T#l).v *> by A27,FINSEQ_2:34
        .= <* (l*T).v *> by A25,A26,FUNCT_4:11
        .= (l*T) * (<*v*>) by A28,FINSEQ_2:34
        .= l * (T * (<*v*>)) by RELAT_1:36
        .= l * (<*T.v*>) by A3,FINSEQ_2:34
        .= <* l.w *> by A5,A6,FINSEQ_2:34;
        then Sum(lCFST((T#l),T,w)) = l.w by A24,RLVECT_1:44;
        hence m.w = l.w by LDef5;
      end;
    end;
    suppose
      A29: not w in Carrier l; then
      A30: l.w = 0.R;
      now
        assume
        A31: m.w <> 0.R;
        then w in Carrier m;
        then consider v being Element of V such that
        A32: v in T"{w} and
        A33: v in Carrier (T#l) by RANKNULL:3,Th36;
        T.v in {w} by A32,FUNCT_1:def 7; then
        A34: T.v = w by TARSKI:def 1;
        A35: Carrier (T#l) c= X by VECTSP_6:def 4;
        T | (Carrier (T#l)) is one-to-one by A1,RANKNULL:2,VECTSP_6:def 4;
        then m.w = (T#l).v by A33,A34,Th37
        .= 0.R by A1,A30,A33,A34,A35,Th42;
        hence contradiction by A31;
      end;
      hence thesis by A29;
    end;
  end;
