
theorem Th50:
  for K be add-associative right_zeroed right_complementable
  distributive non empty doubleLoopStr for V,W be non empty ModuleStr over K
  for f be Functional of V, g be Functional of W holds ker f c= leftker
  FormFunctional(f,g)
proof
  let K be add-associative right_zeroed right_complementable 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);
A1: ker f = {v where v is Vector of V : f.v = 0.K} by VECTSP10:def 9;
  let x be object;
  assume x in ker f;
  then consider v be Vector of V such that
A2: x=v and
A3: f.v=0.K by A1;
  now
    let w be Vector of W;
    thus fg.(v,w) = f.v*g.w by Def10
      .= 0.K by A3;
  end;
  hence thesis by A2;
end;
