let V, W be non empty VectSpStr of 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 *' )

now

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 3
.= ((f . v,w) *' ) + ((g . v,w) *' ) by COMPLFLD:87
.= ((f *' ) . v,w) + ((g . v,w) *' ) by Def8
.= ((f *' ) . v,w) + ((g *' ) . v,w) by Def8
.= ((f *' ) + (g *' )) . v,w by BILINEAR:def 3 ; :: thesis: verum

end;

hence (f + g) *' = (f *' ) + (g *' ) by BINOP_1:2; :: thesis: verum