let K be non empty left-distributive doubleLoopStr ; :: thesis: for V being non empty ModuleStr over K
for r, s being Element of K
for f being Functional of V holds (r + s) * f = (r * f) + (s * f)
let V be non empty ModuleStr over K; :: thesis: for r, s being Element of K
for f being Functional of V holds (r + s) * f = (r * f) + (s * f)
let r, s be Element of K; :: thesis: for f being Functional of V holds (r + s) * f = (r * f) + (s * f)
let f be Functional 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 Def6
.= (r * (f . x)) + (s * (f . x)) by VECTSP_1:def 3
.= ((r * f) . x) + (s * (f . x)) by Def6
.= ((r * f) . x) + ((s * f) . x) by Def6
.= ((r * f) + (s * f)) . x by Def3 ; :: thesis: verum

end;

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