let c1, c2 be complex number ; :: thesis: for g being complex-valued Function holds (g (#) c1) (/) c2 = g (#) (c1 / c2)
let g be complex-valued Function; :: thesis: (g (#) c1) (/) c2 = g (#) (c1 / c2)
( dom (g (#) c1) = dom g & dom ((g (#) c1) (/) c2) = dom (g (#) c1) ) by VALUED_1:def 5;
hence dom ((g (#) c1) (/) c2) = dom (g (#) (c1 / c2)) by VALUED_1:def 5; :: according to FUNCT_1:def 17 :: thesis: for b₁ being set holds
( not b₁ in proj1 ((g (#) c1) (/) c2) or ((g (#) c1) (/) c2) . b₁ = (g (#) (c1 / c2)) . b₁ )
let x be set ; :: thesis: ( not x in proj1 ((g (#) c1) (/) c2) or ((g (#) c1) (/) c2) . x = (g (#) (c1 / c2)) . x )
assume x in dom ((g (#) c1) (/) c2) ; :: thesis: ((g (#) c1) (/) c2) . x = (g (#) (c1 / c2)) . x
thus ((g (#) c1) (/) c2) . x = ((g (#) c1) . x) * (c2 ") by VALUED_1:6
.= ((g . x) * c1) * (c2 ") by VALUED_1:6
.= (g . x) * (c1 / c2)
.= (g (#) (c1 / c2)) . x by VALUED_1:6 ; :: thesis: verum