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 being LeftMod of R
  for A being Subset of V, l being Linear_Combination of A,
      T being linear-transformation of V,W,
      v being Element of V st T|A is one-to-one & v in A holds
  ex X being Subset of V
  st X misses A & T"{T.v} = {v} \/ X
  proof
    let R be Ring;
    let V,W be LeftMod of R;
    let A be Subset of V,
        l be Linear_Combination of A,
        T be linear-transformation of V,W,
        v be Element of V such that
    A1: T|A is one-to-one and
    A2: v in A;
    set X = T"{T.v} \ {v};
    A3: X misses A
    proof
      dom T = [#]V by RANKNULL:7; then
      A4: dom (T|A) = A by RELAT_1:62;
      assume X meets A;
      then consider x being object such that
      A5: x in X and
      A6: x in A by XBOOLE_0:3;
      not x in {v} by A5,XBOOLE_0:def 5; then
      A7: x <> v by TARSKI:def 1;
      x in T"{T.v} by A5,XBOOLE_0:def 5;
      then T.x in {T.v} by FUNCT_1:def 7; then
      A8: T.x = T.v by TARSKI:def 1;
      T.x = (T|A).x by A6,FUNCT_1:49;
      then (T|A).v = (T|A).x by A2,A8,FUNCT_1:49;
      hence contradiction by A1,A2,A4,A6,A7,FUNCT_1:def 4;
    end;
    take X;
    {v} c= T"{T.v}
    proof
      let x be object;
      assume x in {v}; then
      A9: x = v by TARSKI:def 1;
      dom T = [#]V & T.v in {T.v} by TARSKI:def 1,RANKNULL:7;
      hence thesis by A9,FUNCT_1:def 7;
    end;
    hence thesis by A3,XBOOLE_1:45;
  end;
