let R be Ring; for V, U, W being LeftMod of R
for f being linear-transformation of V,U
for g being linear-transformation of U,W holds the carrier of (ker (g * f)) = f " the carrier of (ker g)
let V, U, W be LeftMod of R; for f being linear-transformation of V,U
for g being linear-transformation of U,W holds the carrier of (ker (g * f)) = f " the carrier of (ker g)
let f be linear-transformation of V,U; for g being linear-transformation of U,W holds the carrier of (ker (g * f)) = f " the carrier of (ker g)
let g be linear-transformation of U,W; the carrier of (ker (g * f)) = f " the carrier of (ker g)
thus the carrier of (ker (g * f)) =
(g * f) " {(0. W)}
by LMFirst2
.=
f " (g " {(0. W)})
by RELAT_1:146
.=
f " the carrier of (ker g)
by LMFirst2
; verum