let R be Ring; :: thesis: for V being LeftMod of R
for L1, L2 being Linear_Combination of V holds (vector ((LC_Z_Module V),L1)) - (vector ((LC_Z_Module V),L2)) = L1 - L2
let V be LeftMod of R; :: thesis: for L1, L2 being Linear_Combination of V holds (vector ((LC_Z_Module V),L1)) - (vector ((LC_Z_Module V),L2)) = L1 - L2
let L1, L2 be Linear_Combination of V; :: thesis: (vector ((LC_Z_Module V),L1)) - (vector ((LC_Z_Module V),L2)) = L1 - L2
- L2 in LinComb V by Def29;
then A1: - L2 in LC_Z_Module V ;
- (vector ((LC_Z_Module V),L2)) = - L2 by Th49
.= vector ((LC_Z_Module V),(- L2)) by A1, RLVECT_2:def 1 ;
hence (vector ((LC_Z_Module V),L1)) - (vector ((LC_Z_Module V),L2)) = L1 - L2 by Th47; :: thesis: verum