 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve s for FinSequence,
   V1,V2,V3 for finite-rank free Z_Module,
   f,f1,f2 for Function of V1,V2,
   g for Function of V2,V3,
   b1 for OrdBasis of V1,
   b2 for OrdBasis of V2,
   b3 for OrdBasis of V3,
   v1,v2 for Vector of V2,
   v,w for Element of V1;
 reserve p2,F for FinSequence of V1,
   p1,d for FinSequence of INT.Ring,
   KL for Linear_Combination of V1;

theorem VS10Th1:
  for a being Element of INT.Ring, V being add-associative right_zeroed
  right_complementable vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty ModuleStr over INT.Ring,
      v being Vector of V
  holds (0.INT.Ring)*v = 0.V & a*(0.V) = 0.V
  proof
    let x be Element of INT.Ring;
    let V be add-associative right_zeroed right_complementable
    vector-distributive scalar-distributive scalar-associative scalar-unital
    non empty ModuleStr over INT.Ring, v be Vector of V;
    A1: v+(0.INT.Ring)*v = 1.INT.Ring*v + (0.INT.Ring)*v by VECTSP_1:def 17
    .= (1.INT.Ring+0.INT.Ring)*v by VECTSP_1:def 15
    .= v + 0.V by VECTSP_1:def 17;
    hence (0.INT.Ring)*v = 0.V by RLVECT_1:8;
    hence x*(0.V) = (x*0.INT.Ring)*v by VECTSP_1:def 16
    .= 0.V by A1,RLVECT_1:8;
  end;
