let x be object ; :: thesis: for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds
( x in v + W iff ex u being Vector of V st
( u in W & x = v + u ) )
let R be Ring; :: thesis: for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds
( x in v + W iff ex u being Vector of V st
( u in W & x = v + u ) )
let V be RightMod of R; :: thesis: for v being Vector of V
for W being Submodule of V holds
( x in v + W iff ex u being Vector of V st
( u in W & x = v + u ) )
let v be Vector of V; :: thesis: for W being Submodule of V holds
( x in v + W iff ex u being Vector of V st
( u in W & x = v + u ) )
let W be Submodule of V; :: thesis: ( x in v + W iff ex u being Vector of V st
( u in W & x = v + u ) )
thus ( x in v + W implies ex u being Vector of V st
( u in W & x = v + u ) ) :: thesis: ( ex u being Vector of V st
( u in W & x = v + u ) implies x in v + W )

proof

assume x in v + W ; :: thesis: ex u being Vector of V st
( u in W & x = v + u )
then consider u being Vector 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 Vector 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