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;
