let K be non empty left_unital doubleLoopStr ; :: thesis: for V being non empty ModuleStr over K
for f being Functional of V holds (1. K) * f = f
let V be non empty ModuleStr over K; :: thesis: for f being Functional of V holds (1. K) * f = f
let f be Functional of V; :: thesis: (1. K) * f = f

now :: thesis: for x being Element of V holds ((1. K) * f) . x = f . x

let x be Element of V; :: thesis: ((1. K) * f) . x = f . x
thus ((1. K) * f) . x = (1. K) * (f . x) by Def6
.= f . x ; :: thesis: verum

end;

hence (1. K) * f = f by FUNCT_2:63; :: thesis: verum