let V be non empty ModuleStr over INT.Ring ; :: thesis: for r, s being Element of F_Real
for f being FrFunctional of V holds (r + s) * f = (r * f) + (s * f)
let r, s be Element of F_Real; :: thesis: for f being FrFunctional of V holds (r + s) * f = (r * f) + (s * f)
let f be FrFunctional of V; :: thesis: (r + s) * f = (r * f) + (s * f)

now :: thesis: for x being Element of V holds ((r + s) * f) . x = ((r * f) + (s * f)) . x

let x be Element of V; :: thesis: ((r + s) * f) . x = ((r * f) + (s * f)) . x
thus ((r + s) * f) . x = (r + s) * (f . x) by HDef6
.= (r * (f . x)) + (s * (f . x))
.= ((r * f) . x) + (s * (f . x)) by HDef6
.= ((r * f) . x) + ((s * f) . x) by HDef6
.= ((r * f) + (s * f)) . x by HDef3 ; :: thesis: verum

end;

hence (r + s) * f = (r * f) + (s * f) by FUNCT_2:63; :: thesis: verum