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 Th10:
  (r*s)* AR = r * (s*AR)
  proof
    set rs=r*s;
    hereby let x be object;
      assume x in rs*AR;
      then consider v be Element of R such that
      A1: x=rs*v & v in AR;
      s*v in s*AR & x=r*(s*v) by A1,RLVECT_1:def 7;
      hence x in r*(s*AR);
    end;
    let x be object;
    assume x in r*(s*AR);
    then consider v be Element of R such that
    A2: x=r*v and
    A3: v in s*AR;
    consider w be Element of R such that
    A4: v=s*w and
    A5: w in AR by A3;
    rs*w=x by A2,A4,RLVECT_1:def 7;
    hence thesis by A5;
  end;
