reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;

theorem Th6:
  for V be Abelian right_zeroed non empty addLoopStr for A be Subset of V
    holds 0.V + A = A
 proof
   let V be Abelian right_zeroed non empty addLoopStr;
   let A be Subset of V;
   thus 0.V+A c=A
   proof
     let x be object;
     assume x in 0.V+A;
     then ex s be Element of V st x=0.V+s & s in A;
     hence thesis by RLVECT_1:def 4;
   end;
   let x be object;
   assume A1: x in A;
   then reconsider v=x as Element of V;
   0.V+v=v by RLVECT_1:def 4;
   hence thesis by A1;
 end;
