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 Th34:
  for A being Subset of V, l being Linear_Combination of A, 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 A be Subset of V, l be Linear_Combination of A, 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 Th7;
    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 thesis by A1,A2,A6,A7,A4;
  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 Th7,TARSKI:def 1;
    hence thesis by A9,FUNCT_1:def 7;
  end;
  hence thesis by A3,XBOOLE_1:45;
end;
