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

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

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

now :: thesis: for v being Vector of V

for w being Vector of W holds ((f + g) *') . (v,w) = ((f *') + (g *')) . (v,w)

hence
(f + g) *' = (f *') + (g *')
; :: thesis: verumfor w being Vector of W holds ((f + g) *') . (v,w) = ((f *') + (g *')) . (v,w)

let v be Vector of V; :: thesis: for w being Vector of W holds ((f + g) *') . (v,w) = ((f *') + (g *')) . (v,w)

let w be Vector of W; :: thesis: ((f + g) *') . (v,w) = ((f *') + (g *')) . (v,w)

thus ((f + g) *') . (v,w) = ((f + g) . (v,w)) *' by Def8

.= ((f . (v,w)) + (g . (v,w))) *' by BILINEAR:def 2

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

.= ((f *') . (v,w)) + ((g . (v,w)) *') by Def8

.= ((f *') . (v,w)) + ((g *') . (v,w)) by Def8

.= ((f *') + (g *')) . (v,w) by BILINEAR:def 2 ; :: thesis: verum

end;let w be Vector of W; :: thesis: ((f + g) *') . (v,w) = ((f *') + (g *')) . (v,w)

thus ((f + g) *') . (v,w) = ((f + g) . (v,w)) *' by Def8

.= ((f . (v,w)) + (g . (v,w))) *' by BILINEAR:def 2

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

.= ((f *') . (v,w)) + ((g . (v,w)) *') by Def8

.= ((f *') . (v,w)) + ((g *') . (v,w)) by Def8

.= ((f *') + (g *')) . (v,w) by BILINEAR:def 2 ; :: thesis: verum