
theorem
  for K be add-associative right_zeroed right_complementable
right-distributive non empty doubleLoopStr for V be non empty ModuleStr over
  K for f be alternating bilinear-Form of V,V holds leftker f = rightker f
proof
  let K be add-associative right_zeroed right_complementable
right-distributive non empty doubleLoopStr, V be non empty ModuleStr over K,
  f be alternating bilinear-Form of V,V;
  thus leftker f c= rightker f
  proof
    let x be object;
    assume x in leftker f;
    then consider v be Vector of V such that
A1: v = x and
A2: for w be Vector of V holds f.(v,w)=0.K;
    now
      let w be Vector of V;
      thus f.(w,v) = -f.(v,w) by Th58
        .= -0.K by A2
        .= 0.K by RLVECT_1:12;
    end;
    hence thesis by A1;
  end;
  let x be object;
  assume x in rightker f;
  then consider w be Vector of V such that
A3: w = x and
A4: for v be Vector of V holds f.(v,w)=0.K;
  now
    let v be Vector of V;
    thus f.(w,v) = -f.(v,w) by Th58
      .= -0.K by A4
      .= 0.K by RLVECT_1:12;
  end;
  hence thesis by A3;
end;
