let V be Z_Module; :: thesis: for u, v being VECTOR of V
for W being Submodule of V holds
( u in W iff v + W = (v - u) + W )
let u, v be VECTOR of V; :: thesis: for W being Submodule of V holds
( u in W iff v + W = (v - u) + W )
let W be Submodule of V; :: thesis: ( u in W iff v + W = (v - u) + W )
A1: ( - u in W implies u in W )

proof

assume - u in W ; :: thesis: u in W
then - (- u) in W by Th38;
hence u in W by RLVECT_1:17; :: thesis: verum

end;

( - u in W iff v + W = (v + (- u)) + W ) by Th65;
hence ( u in W iff v + W = (v - u) + W ) by A1, Th38; :: thesis: verum