
theorem
  for L being non empty multLoopStr, P being non empty Subset of L, A
  being LeftLinearCombination of P, i being Element of NAT holds A/^i is
  LeftLinearCombination of P
proof
  let L be non empty multLoopStr, P be non empty Subset of L, A be
  LeftLinearCombination of P, j be Element of NAT;
  set C = A/^j;
  reconsider C as FinSequence of the carrier of L;
  now
    per cases;
    case
A1:   j <= len A;
      then reconsider m = len A - j as Element of NAT by INT_1:5;
      now
        let i be set;
        assume
A2:     i in dom C;
        then reconsider k = i as Element of NAT;
A3:     dom C = Seg(len C) by FINSEQ_1:def 3
          .= Seg(m) by A1,RFINSEQ:def 1;
        then k <= len A - j by A2,FINSEQ_1:1;
        then
A4:     k + j <= (len A + -j) + j by XREAL_1:6;
A5:     k <= k + j by NAT_1:11;
        1 <= k by A2,A3,FINSEQ_1:1;
        then 1 <= k + j by A5,XXREAL_0:2;
        then j + k in Seg(len A) by A4,FINSEQ_1:1;
        then j + k in dom A by FINSEQ_1:def 3;
        then ex u being Element of L, a being Element of P st A/.(j+k) = u * a
        by IDEAL_1:def 9;
        hence ex u being Element of L, a being Element of P st C/.i = u * a by
A2,FINSEQ_5:27;
      end;
      hence thesis by IDEAL_1:def 9;
    end;
    case
      not j <= len A;
      then C = <*>(the carrier of L) by RFINSEQ:def 1;
      then for i being set st i in dom C ex u being Element of L, a being
      Element of P st C/.i = u * a;
      hence thesis by IDEAL_1:def 9;
    end;
  end;
  hence thesis;
end;
