let V be Z_Module; :: thesis: for v being VECTOR of V
for W being Submodule of V
for w being VECTOR of W st w = v holds
- v = - w
let v be VECTOR of V; :: thesis: for W being Submodule of V
for w being VECTOR of W st w = v holds
- v = - w
let W be Submodule of V; :: thesis: for w being VECTOR of W st w = v holds
- v = - w
let w be VECTOR of W; :: thesis: ( w = v implies - v = - w )
A1: ( - v = (- 1) * v & - w = (- 1) * w ) by Th2;
assume w = v ; :: thesis: - v = - w
hence - v = - w by A1, Th29; :: thesis: verum