
theorem hom4a:
for R being add-associative right_zeroed right_complementable
            non empty doubleLoopStr,
    S being add-associative right_zeroed right_complementable
            right-distributive non empty doubleLoopStr,
    f being additive Function of R,S
for x being Element of R
holds f.(-x) = - f.x
proof
let R be add-associative right_zeroed right_complementable
         non empty doubleLoopStr,
    S be add-associative right_zeroed right_complementable
         right-distributive non empty doubleLoopStr,
    f being additive Function of R,S;
let x be Element of R;
0.S = f.(0.R) by hom1
   .= f.(-x + x) by RLVECT_1:5
   .= f.(-x) + f.x by VECTSP_1:def 20;
hence thesis by RLVECT_1:6;
end;
