
theorem LT1:
for R being commutative Ring
for U,V being VectSp of R
for f,g being linear-transformation of U,V
holds f '+' g is linear-transformation of U,V
proof
let R be commutative Ring, U,V be VectSp of R,
    f,g be linear-transformation of U,V;
A: now let x,y be Element of U;
  thus (f '+' g).(x+y)
     = f.(x+y) + g.(x+y) by defp
    .= (f.x + f.y) + g.(x+y) by VECTSP_1:def 20
    .= (f.x + f.y) + (g.x + g.y) by VECTSP_1:def 20
    .= f.x + (f.y + (g.x + g.y)) by RLVECT_1:def 3
    .= f.x + ((f.y + g.y) + g.x) by RLVECT_1:def 3
    .= (f.x + g.x) + (f.y + g.y) by RLVECT_1:def 3
    .= (f '+' g).x + (f.y + g.y) by defp
    .= (f '+' g).x + (f '+' g).y by defp;
  end;
now let a be Element of R, x be Element of U;
  thus (f '+' g).(a*x)
     = f.(a*x) + g.(a*x) by defp
    .= (a * f.x) + g.(a*x) by MOD_2:def 2
    .= (a * f.x) + (a * g.x) by MOD_2:def 2
    .= a * (f.x + g.x) by VECTSP_1:def 14
    .= a * ((f '+' g).x) by defp;
   end;
hence thesis by A,MOD_2:def 2,VECTSP_1:def 20;
end;
