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;

theorem Th13:
  for F being Ring, V, W being VectSp of F,
      T being linear-transformation of V,W holds
  for y being Element of W holds y in im T iff
  ex x being Element of V st y = T.x
proof
  let F be Ring, V, W be VectSp of F,
      T be linear-transformation of V,W;
  let y be Element of W;
A1: y in im T implies ex x being Element of V st y = T.x
  proof
    assume y in im T;
    then reconsider y as Element of im T;
    [#](im T) = T .: [#]V by Def2;
    then consider x being object such that
    x in dom T and
A2: x in [#]V and
A3: y = T.x by FUNCT_1:def 6;
    reconsider x as Element of V by A2;
    take x;
    thus thesis by A3;
  end;
  (ex x being Element of V st y = T.x) implies y in im T
  proof
    assume
A4: ex x being Element of V st y = T.x;
    dom T = [#]V by Th7;
    then y in T .: [#]V by A4,FUNCT_1:def 6;
    then y in [#](im T) by Def2;
    hence thesis;
  end;
  hence thesis by A1;
end;
