
theorem Th51:
  for K be add-associative right_zeroed right_complementable
  associative commutative well-unital almost_left_invertible distributive non
empty doubleLoopStr for V, W be non empty ModuleStr over K for f be Functional
of V, g be Functional of W st g <> 0Functional(W) holds leftker FormFunctional(
  f,g) = ker f
proof
  let K be add-associative right_zeroed right_complementable associative
  commutative well-unital almost_left_invertible distributive non empty
doubleLoopStr, V, W be non empty ModuleStr over K, f be Functional of V, g be
  Functional of W;
  set fg = FormFunctional(f,g);
  assume
A1: g <> 0Functional(W);
A2: ker f = {v where v is Vector of V : f.v = 0.K} by VECTSP10:def 9;
  thus leftker fg c= ker f
  proof
    let x be object;
    assume x in leftker fg;
    then consider v be Vector of V such that
A3: x=v and
A4: for w be Vector of W holds fg.(v,w) = 0.K;
    assume not x in ker f;
    then
A5: f.v <> 0.K by A2,A3;
    now
      let w be Vector of W;
      f.v*g.w = fg.(v,w) by Def10
        .= 0.K by A4;
      hence g.w = 0.K by A5,VECTSP_1:12
        .= (0Functional(W)).w by HAHNBAN1:14;
    end;
    hence contradiction by A1,FUNCT_2:63;
  end;
  thus thesis by Th50;
end;
