 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;

theorem
  for cf,cf1 be complex-valued FinSubsequence
    for P be Permutation of dom cf st cf1 = cf * P
  holds Sum Seq(cf|X) = Sum Seq(cf1|P"X)
proof
  let cf,cf1 be complex-valued FinSubsequence;
  rng Seq(cf|X)c=COMPLEX by VALUED_0:def 1;
  then reconsider SfX=Seq(cf|X) as FinSequence of COMPLEX by FINSEQ_1:def 4;
  let P be Permutation of dom cf such that
   A1: cf1=cf*P;
  consider Q be Permutation of dom Seq(cf1|P"X) such that
   A2: Seq(cf|X)=Seq(cf1|P"X)*Q by A1,Th9;
  rng Seq(cf1|P"X)c=COMPLEX by VALUED_0:def 1;
  then reconsider SfPX=Seq(cf1|P"X) as FinSequence of COMPLEX
    by FINSEQ_1:def 4;
  thus Sum Seq(cf1|P"X)=addcomplex"**"SfPX by RVSUM_1:def 11
   .=addcomplex"**"SfX by A2,FINSOP_1:7
   .=Sum Seq(cf|X) by RVSUM_1:def 11;
end;
