let x be object ; :: thesis: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for v being Element of V
for W being Subspace of V holds
( x in v + W iff ex u being Element of V st
( u in W & x = v + u ) )

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 v being Element of V
for W being Subspace of V holds
( x in v + W iff ex u being Element of V st
( u in W & x = v + u ) )

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 v being Element of V
for W being Subspace of V holds
( x in v + W iff ex u being Element of V st
( u in W & x = v + u ) )

let v be Element of V; :: thesis: for W being Subspace of V holds
( x in v + W iff ex u being Element of V st
( u in W & x = v + u ) )

let W be Subspace of V; :: thesis: ( x in v + W iff ex u being Element of V st
( u in W & x = v + u ) )

thus ( x in v + W implies ex u being Element of V st
( u in W & x = v + u ) ) :: thesis: ( ex u being Element of V st
( u in W & x = v + u ) implies x in v + W )
proof
assume x in v + W ; :: thesis: ex u being Element of V st
( u in W & x = v + u )

then consider u being Element of V such that
A1: ( x = v + u & u in W ) ;
take u ; :: thesis: ( u in W & x = v + u )
thus ( u in W & x = v + u ) by A1; :: thesis: verum
end;
given u being Element of V such that A2: ( u in W & x = v + u ) ; :: thesis: x in v + W
thus x in v + W by A2; :: thesis: verum