let V be Z_Module; :: thesis: for u, v being VECTOR of V
for W being Submodule of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
let u, v be VECTOR of V; :: thesis: for W being Submodule of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
let W be Submodule of V; :: thesis: ( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
thus ( u in v + W implies ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) ) :: thesis: ( ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) implies u in v + W )

proof

assume u in v + W ; :: thesis: ex v1 being VECTOR of V st
( v1 in W & u = v + v1 )
then ex v1 being VECTOR of V st
( u = v + v1 & v1 in W ) ;
hence ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) ; :: thesis: verum

end;

given v1 being VECTOR of V such that A1: ( v1 in W & u = v + v1 ) ; :: thesis: u in v + W
thus u in v + W by A1; :: thesis: verum