
theorem hom2:
for R being right_unital non empty doubleLoopStr,
    S being add-associative right_zeroed right_complementable right_unital
            Abelian right-distributive domRing-like non empty doubleLoopStr,
    f being multiplicative Function of R,S
holds f.(1.R) = 0.S or f.(1.R) = 1.S
proof
let R be right_unital non empty doubleLoopStr,
    S be add-associative right_zeroed right_complementable Abelian
         right_unital right-distributive domRing-like non empty doubleLoopStr,
    f be multiplicative Function of R,S;
A: f.(1.R) = f.(1.R * 1.R)
          .= f.(1.R) * f.(1.R) by GROUP_6:def 6;
B: f.(1.R) * (1.S - f.(1.R))
     = f.(1.R) * 1.S + f.(1.R) * (-f.(1.R)) by VECTSP_1:def 2
    .= f.(1.R) * 1.S + -(f.(1.R) * f.(1.R)) by VECTSP_1:8
    .= f.(1.R) - f.(1.R) * f.(1.R)
    .= 0.S by A,RLVECT_1:15;
now assume C: f.(1.R) <> 0.S;
  thus f.(1.R) = f.(1.R) + 0.S
              .= f.(1.R) + (1.S + -f.(1.R)) by C,B,VECTSP_2:def 1
              .= (f.(1.R) + -f.(1.R)) + 1.S by RLVECT_1:def 3
              .= 0.S + 1.S by RLVECT_1:5
              .= 1.S;
  end;
hence thesis;
end;
