let K be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr ; :: thesis: for V, W being non empty ModuleStr over K

for f being Functional of V

for v being Vector of V

for w being Vector of W holds (FormFunctional (f,(0Functional W))) . (v,w) = 0. K

let V, W be non empty ModuleStr over K; :: thesis: for f being Functional of V

for v being Vector of V

for w being Vector of W holds (FormFunctional (f,(0Functional W))) . (v,w) = 0. K

let f be Functional of V; :: thesis: for v being Vector of V

for w being Vector of W holds (FormFunctional (f,(0Functional W))) . (v,w) = 0. K

let v be Vector of V; :: thesis: for w being Vector of W holds (FormFunctional (f,(0Functional W))) . (v,w) = 0. K

let y be Vector of W; :: thesis: (FormFunctional (f,(0Functional W))) . (v,y) = 0. K

set 0F = 0Functional W;

set F = FormFunctional (f,(0Functional W));

thus (FormFunctional (f,(0Functional W))) . (v,y) = (f . v) * ((0Functional W) . y) by Def10

.= (f . v) * (0. K) by FUNCOP_1:7

.= 0. K ; :: thesis: verum

for f being Functional of V

for v being Vector of V

for w being Vector of W holds (FormFunctional (f,(0Functional W))) . (v,w) = 0. K

let V, W be non empty ModuleStr over K; :: thesis: for f being Functional of V

for v being Vector of V

for w being Vector of W holds (FormFunctional (f,(0Functional W))) . (v,w) = 0. K

let f be Functional of V; :: thesis: for v being Vector of V

for w being Vector of W holds (FormFunctional (f,(0Functional W))) . (v,w) = 0. K

let v be Vector of V; :: thesis: for w being Vector of W holds (FormFunctional (f,(0Functional W))) . (v,w) = 0. K

let y be Vector of W; :: thesis: (FormFunctional (f,(0Functional W))) . (v,y) = 0. K

set 0F = 0Functional W;

set F = FormFunctional (f,(0Functional W));

thus (FormFunctional (f,(0Functional W))) . (v,y) = (f . v) * ((0Functional W) . y) by Def10

.= (f . v) * (0. K) by FUNCOP_1:7

.= 0. K ; :: thesis: verum