let V be Z_Module; :: thesis: for v being VECTOR of V
for a being Integer
for W being Submodule of V
for w being VECTOR of W st w = v holds
a * w = a * v
let v be VECTOR of V; :: thesis: for a being Integer
for W being Submodule of V
for w being VECTOR of W st w = v holds
a * w = a * v
let a be Integer; :: thesis: for W being Submodule of V
for w being VECTOR of W st w = v holds
a * w = a * v
let W be Submodule of V; :: thesis: for w being VECTOR of W st w = v holds
a * w = a * v
let w be VECTOR of W; :: thesis: ( w = v implies a * w = a * v )
reconsider a = a as Element of INT by INT_1:def 2;
assume A1: w = v ; :: thesis: a * w = a * v
a * w = ( the Mult of V | [:INT, the carrier of W:]) . [a,w] by Def9;
hence a * w = a * v by A1, FUNCT_1:49; :: thesis: verum