
theorem LT2:
for R being commutative Ring
for U,V being VectSp of R
for f being linear-transformation of U,V
for a being Element of R
holds a '*' f is linear-transformation of U,V
proof
let R be commutative Ring, U,V be VectSp of R,
    f be linear-transformation of U,V, a be Element of R;
A: now let x,y be Element of U;
  thus (a '*' f).(x+y)
     = a * f.(x+y) by defmu
    .= a * (f.x + f.y) by VECTSP_1:def 20
    .= (a * f.x) + (a * f.y) by VECTSP_1:def 14
    .= (a '*' f).x + (a * f.y) by defmu
    .= (a '*' f).x + (a '*' f).y by defmu;
  end;
now let b be Element of R, x be Element of U;
  thus (a '*' f).(b*x)
     = a * f.(b*x) by defmu
    .= a * (b * f.x) by MOD_2:def 2
    .= (a * b) * f.x by VECTSP_1:def 16
    .= b * (a * f.x) by VECTSP_1:def 16
    .= b * ((a '*' f).x) by defmu;
   end;
hence thesis by A,MOD_2:def 2,VECTSP_1:def 20;
end;
