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 )
proof
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 ; :: thesis: u = v - x
thus u = v - x by ; :: thesis: verum
end;
given v1 being Element of V such that A3: v1 in W and
A4: u = v - v1 ; :: thesis: u in v + W
- v1 in W by ;
hence u in v + W by A4; :: thesis: verum