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

for f being Functional of V

for g being Functional of W st g <> 0Functional W holds

leftker (FormFunctional (f,g)) = ker f

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

for g being Functional of W st g <> 0Functional W holds

leftker (FormFunctional (f,g)) = ker f

let f be Functional of V; :: thesis: for g being Functional of W st g <> 0Functional W holds

leftker (FormFunctional (f,g)) = ker f

let g be Functional of W; :: thesis: ( g <> 0Functional W implies leftker (FormFunctional (f,g)) = ker f )

set fg = FormFunctional (f,g);

assume A1: g <> 0Functional W ; :: thesis: leftker (FormFunctional (f,g)) = ker f

A2: ker f = { v where v is Vector of V : f . v = 0. K } by VECTSP10:def 9;

thus leftker (FormFunctional (f,g)) c= ker f :: according to XBOOLE_0:def 10 :: thesis: ker f c= leftker (FormFunctional (f,g))

for f being Functional of V

for g being Functional of W st g <> 0Functional W holds

leftker (FormFunctional (f,g)) = ker f

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

for g being Functional of W st g <> 0Functional W holds

leftker (FormFunctional (f,g)) = ker f

let f be Functional of V; :: thesis: for g being Functional of W st g <> 0Functional W holds

leftker (FormFunctional (f,g)) = ker f

let g be Functional of W; :: thesis: ( g <> 0Functional W implies leftker (FormFunctional (f,g)) = ker f )

set fg = FormFunctional (f,g);

assume A1: g <> 0Functional W ; :: thesis: leftker (FormFunctional (f,g)) = ker f

A2: ker f = { v where v is Vector of V : f . v = 0. K } by VECTSP10:def 9;

thus leftker (FormFunctional (f,g)) c= ker f :: according to XBOOLE_0:def 10 :: thesis: ker f c= leftker (FormFunctional (f,g))

proof

thus
ker f c= leftker (FormFunctional (f,g))
by Th50; :: thesis: verum
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in leftker (FormFunctional (f,g)) or x in ker f )

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

then consider v being Vector of V such that

A3: x = v and

A4: for w being Vector of W holds (FormFunctional (f,g)) . (v,w) = 0. K ;

assume not x in ker f ; :: thesis: contradiction

then A5: f . v <> 0. K by A2, A3;

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

then consider v being Vector of V such that

A3: x = v and

A4: for w being Vector of W holds (FormFunctional (f,g)) . (v,w) = 0. K ;

assume not x in ker f ; :: thesis: contradiction

then A5: f . v <> 0. K by A2, A3;

now :: thesis: for w being Vector of W holds g . w = (0Functional W) . w

hence
contradiction
by A1, FUNCT_2:63; :: thesis: verumlet w be Vector of W; :: thesis: g . w = (0Functional W) . w

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

.= 0. K by A4 ;

hence g . w = 0. K by A5, VECTSP_1:12

.= (0Functional W) . w by HAHNBAN1:14 ;

:: thesis: verum

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

.= 0. K by A4 ;

hence g . w = 0. K by A5, VECTSP_1:12

.= (0Functional W) . w by HAHNBAN1:14 ;

:: thesis: verum