let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for u, v being Element of V

for W being Subspace of V holds

( u in v + W iff ex v1 being Element of V st

( v1 in W & u = v - v1 ) )

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for u, v being Element of V

for W being Subspace of V holds

( u in v + W iff ex v1 being Element of V st

( v1 in W & u = v - v1 ) )

let u, v be Element of V; :: thesis: for W being Subspace of V holds

( u in v + W iff ex v1 being Element of V st

( v1 in W & u = v - v1 ) )

let W be Subspace of V; :: thesis: ( u in v + W iff ex v1 being Element of V st

( v1 in W & u = v - v1 ) )

thus ( u in v + W implies ex v1 being Element of V st

( v1 in W & u = v - v1 ) ) :: thesis: ( ex v1 being Element of V st

( v1 in W & u = v - v1 ) implies u in v + W )

A4: u = v - v1 ; :: thesis: u in v + W

- v1 in W by A3, Th22;

hence u in v + W by A4; :: thesis: verum

for u, v being Element of V

for W being Subspace of V holds

( u in v + W iff ex v1 being Element of V st

( v1 in W & u = v - v1 ) )

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for u, v being Element of V

for W being Subspace of V holds

( u in v + W iff ex v1 being Element of V st

( v1 in W & u = v - v1 ) )

let u, v be Element of V; :: thesis: for W being Subspace of V holds

( u in v + W iff ex v1 being Element of V st

( v1 in W & u = v - v1 ) )

let W be Subspace of V; :: thesis: ( u in v + W iff ex v1 being Element of V st

( v1 in W & u = v - v1 ) )

thus ( u in v + W implies ex v1 being Element of V st

( v1 in W & u = v - v1 ) ) :: thesis: ( ex v1 being Element of V st

( v1 in W & u = v - v1 ) implies u in v + W )

proof

given v1 being Element of V such that A3:
v1 in W
and
assume
u in v + W
; :: thesis: ex v1 being Element of V st

( v1 in W & u = v - v1 )

then consider v1 being Element of V such that

A1: u = v + v1 and

A2: v1 in W ;

take x = - v1; :: thesis: ( x in W & u = v - x )

thus x in W by A2, Th22; :: thesis: u = v - x

thus u = v - x by A1, RLVECT_1:17; :: thesis: verum

end;( v1 in W & u = v - v1 )

then consider v1 being Element of V such that

A1: u = v + v1 and

A2: v1 in W ;

take x = - v1; :: thesis: ( x in W & u = v - x )

thus x in W by A2, Th22; :: thesis: u = v - x

thus u = v - x by A1, RLVECT_1:17; :: thesis: verum

A4: u = v - v1 ; :: thesis: u in v + W

- v1 in W by A3, Th22;

hence u in v + W by A4; :: thesis: verum