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 Th11:
  1 * AR = AR
  proof
    hereby let x be object;
      assume x in 1*AR;
      then ex v be Element of R st x=1*v & v in AR;
      hence x in AR by RLVECT_1:def 8;
    end;
    let x be object such that
    A1: x in AR;
    reconsider v=x as Element of R by A1;
    x=1*v by RLVECT_1:def 8;
    hence thesis by A1;
  end;
