
theorem Th14:
  for n being Ordinal, L being left_zeroed right_add-cancelable
  right-distributive non empty doubleLoopStr, p being Series of n,L, b being
  bag of n, a being Element of L holds b *' (a * p) = a * (b *' p)
proof
  let n be Ordinal, L be left_zeroed right_add-cancelable right-distributive
non empty doubleLoopStr, p be Series of n,L, b be bag of n, a be Element of L;
  set q1 = b *' (a * p), q2 = a * (b *' p);
A1: now
    let x be object;
    assume x in dom q1;
    then reconsider u = x as bag of n;
    now
      per cases;
      case
A2:     b divides u;
        hence q1.u = (a*p).(u-'b) by POLYRED:def 1
          .= a * p.(u-'b) by POLYNOM7:def 9
          .= a * (b*'p).u by A2,POLYRED:def 1
          .= q2.u by POLYNOM7:def 9;
      end;
      case
A3:     not b divides u;
        hence q1.u = 0.L by POLYRED:def 1
          .= a * 0.L by BINOM:2
          .= a * (b*'p).u by A3,POLYRED:def 1
          .= q2.u by POLYNOM7:def 9;
      end;
    end;
    hence q1.x = q2.x;
  end;
  dom q1 = Bags n by FUNCT_2:def 1
    .= dom q2 by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
