
theorem Th55:
  for V,W be non empty ModuleStr over F_Complex, f be Form of V,W
  holds leftker f = leftker f*' & rightker f = rightker f*'
proof
  let V,W be non empty ModuleStr over F_Complex, f be Form of V,W;
  set K = F_Complex;
  thus leftker f c= leftker f*'
  proof
    let x be object;
    assume x in leftker f;
    then consider v be Vector of V such that
A1: x=v and
A2: for w be Vector of W holds f.(v,w)= 0.K;
    now
      let w be Vector of W;
      (f.(v,w))*' = 0.K by A2,COMPLFLD:47;
      hence (f*').(v,w) = 0.K by Def8;
    end;
    hence thesis by A1;
  end;
  thus leftker f*' c= leftker f
  proof
    let x be object;
    assume x in leftker f*';
    then consider v be Vector of V such that
A3: x=v and
A4: for w be Vector of W holds f*'.(v,w)= 0.K;
    now
      let w be Vector of W;
      (f*'.(v,w))*' = 0.K by A4,COMPLFLD:47;
      then (f.(v,w))*'*' = 0.K by Def8;
      hence f.(v,w) = 0.K;
    end;
    hence thesis by A3;
  end;
  thus rightker f c= rightker f*'
  proof
    let x be object;
    assume x in rightker f;
    then consider w be Vector of W such that
A5: x=w and
A6: for v be Vector of V holds f.(v,w)= 0.K;
    now
      let v be Vector of V;
      (f.(v,w))*' = 0.K by A6,COMPLFLD:47;
      hence (f*').(v,w) = 0.K by Def8;
    end;
    hence thesis by A5;
  end;
  let x be object;
  assume x in rightker f*';
  then consider w be Vector of W such that
A7: x=w and
A8: for v be Vector of V holds f*'.(v,w)= 0.K;
  now
    let v be Vector of V;
    (f*'.(v,w))*' = 0.K by A8,COMPLFLD:47;
    then (f.(v,w))*'*' = 0.K by Def8;
    hence f.(v,w) = 0.K;
  end;
  hence thesis by A7;
end;
