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))

proof

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;

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

let 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;

hence contradiction by A1, FUNCT_2:63; :: thesis: verum

end;

thus ker f c= leftker (FormFunctional (f,g)) by Th50; :: thesis: verum