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

for f being Functional of V

for g being Functional of W holds ker f c= leftker (FormFunctional (f,g))

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

for g being Functional of W holds ker f c= leftker (FormFunctional (f,g))

let f be Functional of V; :: thesis: for g being Functional of W holds ker f c= leftker (FormFunctional (f,g))

let g be Functional of W; :: thesis: ker f c= leftker (FormFunctional (f,g))

set fg = FormFunctional (f,g);

A1: ker f = { v where v is Vector of V : f . v = 0. K } by VECTSP10:def 9;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in ker f or x in leftker (FormFunctional (f,g)) )

assume x in ker f ; :: thesis: x in leftker (FormFunctional (f,g))

then consider v being Vector of V such that

A2: x = v and

A3: f . v = 0. K by A1;

for f being Functional of V

for g being Functional of W holds ker f c= leftker (FormFunctional (f,g))

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

for g being Functional of W holds ker f c= leftker (FormFunctional (f,g))

let f be Functional of V; :: thesis: for g being Functional of W holds ker f c= leftker (FormFunctional (f,g))

let g be Functional of W; :: thesis: ker f c= leftker (FormFunctional (f,g))

set fg = FormFunctional (f,g);

A1: ker f = { v where v is Vector of V : f . v = 0. K } by VECTSP10:def 9;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in ker f or x in leftker (FormFunctional (f,g)) )

assume x in ker f ; :: thesis: x in leftker (FormFunctional (f,g))

then consider v being Vector of V such that

A2: x = v and

A3: f . v = 0. K by A1;

now :: thesis: for w being Vector of W holds (FormFunctional (f,g)) . (v,w) = 0. K

hence
x in leftker (FormFunctional (f,g))
by A2; :: thesis: verumlet w be Vector of W; :: thesis: (FormFunctional (f,g)) . (v,w) = 0. K

thus (FormFunctional (f,g)) . (v,w) = (f . v) * (g . w) by Def10

.= 0. K by A3 ; :: thesis: verum

end;thus (FormFunctional (f,g)) . (v,w) = (f . v) * (g . w) by Def10

.= 0. K by A3 ; :: thesis: verum