let K be Ring; :: thesis: for V being LeftMod of K

for W being Subspace of V

for x being Element of V .. W ex a being Vector of V st x = a .. W

let V be LeftMod of K; :: thesis: for W being Subspace of V

for x being Element of V .. W ex a being Vector of V st x = a .. W

let W be Subspace of V; :: thesis: for x being Element of V .. W ex a being Vector of V st x = a .. W

let x be Element of V .. W; :: thesis: ex a being Vector of V st x = a .. W

consider a being Vector of V such that

A1: x = a + W by Def12;

take a ; :: thesis: x = a .. W

thus x = a .. W by A1; :: thesis: verum

for W being Subspace of V

for x being Element of V .. W ex a being Vector of V st x = a .. W

let V be LeftMod of K; :: thesis: for W being Subspace of V

for x being Element of V .. W ex a being Vector of V st x = a .. W

let W be Subspace of V; :: thesis: for x being Element of V .. W ex a being Vector of V st x = a .. W

let x be Element of V .. W; :: thesis: ex a being Vector of V st x = a .. W

consider a being Vector of V such that

A1: x = a + W by Def12;

take a ; :: thesis: x = a .. W

thus x = a .. W by A1; :: thesis: verum