let V be non empty ModuleStr over F_Complex ; :: thesis: for f, g being Functional of V holds (f + g) *' = (f *') + (g *')

let f, g be Functional of V; :: thesis: (f + g) *' = (f *') + (g *')

let f, g be Functional of V; :: thesis: (f + g) *' = (f *') + (g *')

now :: thesis: for v being Vector of V holds ((f + g) *') . v = ((f *') + (g *')) . v

hence
(f + g) *' = (f *') + (g *')
by FUNCT_2:63; :: thesis: verumlet v be Vector of V; :: thesis: ((f + g) *') . v = ((f *') + (g *')) . v

thus ((f + g) *') . v = ((f + g) . v) *' by Def2

.= ((f . v) + (g . v)) *' by HAHNBAN1:def 3

.= ((f . v) *') + ((g . v) *') by COMPLFLD:51

.= ((f *') . v) + ((g . v) *') by Def2

.= ((f *') . v) + ((g *') . v) by Def2

.= ((f *') + (g *')) . v by HAHNBAN1:def 3 ; :: thesis: verum

end;thus ((f + g) *') . v = ((f + g) . v) *' by Def2

.= ((f . v) + (g . v)) *' by HAHNBAN1:def 3

.= ((f . v) *') + ((g . v) *') by COMPLFLD:51

.= ((f *') . v) + ((g . v) *') by Def2

.= ((f *') . v) + ((g *') . v) by Def2

.= ((f *') + (g *')) . v by HAHNBAN1:def 3 ; :: thesis: verum