
theorem Th53:
  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 f <> 0Functional(V) holds rightker FormFunctional
  (f,g) = ker g
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: f <> 0Functional(V);
A2: ker g = {w where w is Vector of W : g.w = 0.K} by VECTSP10:def 9;
  thus rightker fg c= ker g
  proof
    let x be object;
    assume x in rightker fg;
    then consider w be Vector of W such that
A3: x=w and
A4: for v be Vector of V holds fg.(v,w) = 0.K;
    assume not x in ker g;
    then
A5: g.w <> 0.K by A2,A3;
    now
      let v be Vector of V;
      f.v*g.w = fg.(v,w) by Def10
        .= 0.K by A4;
      hence f.v = 0.K by A5,VECTSP_1:12
        .= (0Functional(V)).v by HAHNBAN1:14;
    end;
    hence contradiction by A1,FUNCT_2:63;
  end;
  thus thesis by Th52;
end;
