let V be Z_Module; :: thesis: for v being VECTOR of V
for a being Integer
for W being Submodule of V st v in W holds
a * v in v + W
let v be VECTOR of V; :: thesis: for a being Integer
for W being Submodule of V st v in W holds
a * v in v + W
let a be Integer; :: thesis: for W being Submodule of V st v in W holds
a * v in v + W
let W be Submodule of V; :: thesis: ( v in W implies a * v in v + W )
assume v in W ; :: thesis: a * v in v + W
then A1: (a - 1) * v in W by Th37;
a * v = ((a - 1) + 1) * v
.= ((a - 1) * v) + (1 * v) by Def3
.= v + ((a - 1) * v) by Def5 ;
hence a * v in v + W by A1; :: thesis: verum