
theorem Th42:
  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 additiveSAF homogeneousSAF Form of V,W holds leftker 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 additiveSAF homogeneousSAF Form of V,W;
  set V1 = leftker f;
  thus for v,u be Vector of V st v in V1 & u in V1 holds v + u in V1
  proof
    let v,u be Vector of V;
    assume that
A1: v in V1 and
A2: u in V1;
    consider u1 be Vector of V such that
A3: u1= u and
A4: for w be Vector of W holds f.(u1,w)=0.K by A2;
    consider v1 be Vector of V such that
A5: v1= v and
A6: for w be Vector of W holds f.(v1,w)=0.K by A1;
    now
      let w be Vector of W;
      thus f.(v+u,w) = f.(v1,w) + f.(u1,w) by A5,A3,Th26
        .= 0.K + f.(u1,w) 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 V;
  assume v in V1;
  then consider v1 be Vector of V such that
A7: v1= v and
A8: for w be Vector of W holds f.(v1,w)=0.K;
  now
    let w be Vector of W;
    thus f.(a*v,w) = a*f.(v1,w) by A7,Th31
      .= a*0.K by A8
      .= 0.K;
  end;
  hence thesis;
end;
