 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 P be Permutation of dom cf st cf1 = cf*P holds Sum (cf1-X) = Sum(cf-X)
proof
  rng(cf1-X)c=COMPLEX by VALUED_0:def 1;
  then reconsider fPX=cf1-X as FinSequence of COMPLEX by FINSEQ_1:def 4;
  rng(cf-X)c=COMPLEX by VALUED_0:def 1;
  then reconsider fX=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(cf-X) such that
   A2: cf1-X=(cf-X)*Q by A1,Th7;
  thus Sum(cf1-X)=addcomplex"**"fPX by RVSUM_1:def 11
   .=addcomplex"**"fX by A2,FINSOP_1:7
   .=Sum(cf-X) by RVSUM_1:def 11;
end;
