
theorem Th43:
  for K be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr for V,W be non empty ModuleStr
  over K for f be additiveFAF homogeneousFAF Form of V,W holds rightker f is
  linearly-closed
proof
  let K be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr;
  let V,W be non empty ModuleStr over K;
  let f be additiveFAF homogeneousFAF Form of V,W;
  set V1 = rightker f;
  thus for v,u be Vector of W st v in V1 & u in V1 holds v + u in V1
  proof
    let v,u be Vector of W;
    assume that
A1: v in V1 and
A2: u in V1;
    consider u1 be Vector of W such that
A3: u1= u and
A4: for w be Vector of V holds f.(w,u1)=0.K by A2;
    consider v1 be Vector of W such that
A5: v1= v and
A6: for w be Vector of V holds f.(w,v1)=0.K by A1;
    now
      let w be Vector of V;
      thus f.(w,v+u) = f.(w,v1) + f.(w,u1) by A5,A3,Th27
        .= 0.K + f.(w,u1) by A6
        .= 0.K + 0.K by A4
        .= 0.K by RLVECT_1:def 4;
    end;
    hence thesis;
  end;
  let a be Element of K, v be Vector of W;
  assume v in V1;
  then consider v1 be Vector of W such that
A7: v1= v and
A8: for w be Vector of V holds f.(w,v1)=0.K;
  now
    let w be Vector of V;
    thus f.(w,a*v) = a*f.(w,v1) by A7,Th32
      .= a*0.K by A8
      .= 0.K;
  end;
  hence thesis;
end;
