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 Th5:
  for V be add-associative non empty addLoopStr
     for A be Subset of V for v,w be Element of V
    holds (v+w)+A = v+(w+A)
 proof
   let V be add-associative non empty addLoopStr;
   let A be Subset of V;
   let v,w be Element of V;
   set vw=v+w;
   thus vw+A c=v+(w+A)
   proof
     let x be object;
     assume x in vw+A;
     then consider s be Element of V such that
     A1: x=vw+s & s in A;
     w+s in w+A & x=v+(w+s) by A1,RLVECT_1:def 3;
     hence thesis;
   end;
   let x be object;
   assume x in v+(w+A);
   then consider s be Element of V such that
   A2: x=v+s and
   A3: s in w+A;
   consider r be Element of V such that
   A4: s=w+r and
   A5: r in A by A3;
   x=vw+r by A2,A4,RLVECT_1:def 3;
   hence thesis by A5;
 end;
