
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|(Seg j);
  reconsider C as FinSequence of the carrier of L by FINSEQ_1:18;
  now
    let i be set;
A1: dom C c= dom A by RELAT_1:60;
    assume
A2: i in dom C;
    then C.i = A.i by FUNCT_1:47;
    then C/.i = A.i by A2,PARTFUN1:def 6
      .= A/.i by A2,A1,PARTFUN1:def 6;
    hence ex u being Element of L, a being Element of P st C/.i = u * a by A2
,A1,IDEAL_1:def 9;
  end;
  then C is LeftLinearCombination of P by IDEAL_1:def 9;
  hence thesis by FINSEQ_1:def 16;
end;
