let c be complex number ; :: thesis: for g, h being complex-valued Function holds (g /" h) (/) c = g /" (h (#) c)
let g, h be complex-valued Function; :: thesis: (g /" h) (/) c = g /" (h (#) c)
A3: dom (g /" h) = (dom g) /\ (dom h) by VALUED_1:16;
A4: dom (h (#) c) = dom h by VALUED_1:def 5;
dom ((g /" h) (/) c) = dom (g /" h) by VALUED_1:def 5;
hence dom ((g /" h) (/) c) = dom (g /" (h (#) c)) by A3, A4, VALUED_1:16; :: according to FUNCT_1:def 17 :: thesis: for b₁ being set holds
( not b₁ in dom ((g /" h) (/) c) or ((g /" h) (/) c) . b₁ = (g /" (h (#) c)) . b₁ )
let x be set ; :: thesis: ( not x in dom ((g /" h) (/) c) or ((g /" h) (/) c) . x = (g /" (h (#) c)) . x )
assume x in dom ((g /" h) (/) c) ; :: thesis: ((g /" h) (/) c) . x = (g /" (h (#) c)) . x
thus ((g /" h) (/) c) . x = ((g /" h) . x) * (c " ) by VALUED_1:6
.= ((g . x) / (h . x)) / c by VALUED_1:17
.= (g . x) / ((h . x) * c) by XCMPLX_1:79
.= (g . x) / ((h (#) c) . x) by VALUED_1:6
.= (g /" (h (#) c)) . x by VALUED_1:17 ; :: thesis: verum