reserve n,m for Nat;

theorem Th24:
  for f,g being real-valued FinSequence,P being Permutation of dom g
  st f = g*P & len g>=1 holds -f=(-g)*P
proof
  let f,g be real-valued FinSequence,P be Permutation of dom g;
  assume that
A1: f = g*P and
A2: len g>=1;
A3: rng P=dom g by FUNCT_2:def 3;
A4: dom (-g)=dom g by VALUED_1:8;
  then
A5: rng ((-g)*P) = rng (-g) by A3,RELAT_1:28;
A6: dom (-f)=dom (g*P) by A1,VALUED_1:8;
  then
A7: dom (-f)=dom P by A3,RELAT_1:27;
  then
A8: dom (-f)=dom ((-g)*P) by A3,A4,RELAT_1:27;
A9: dom g=Seg len g by FINSEQ_1:def 3;
  then dom P=dom g by A2,FUNCT_2:def 1;
  then (-g)*P is FinSequence by A9,A7,A8,FINSEQ_1:def 2;
  then reconsider k=(-g)*P as FinSequence of REAL by A5,FINSEQ_1:def 4;
  for i being Nat st i in dom (-f) holds (-f).i=k.i
  proof
    let i be Nat;
    assume
A10: i in dom (-f);
    reconsider j=P.i as Nat;
    (-f).i=-(f.i) by RVSUM_1:17
      .=-(g.(P.i)) by A1,A6,A10,FUNCT_1:12
      .=(-g).(j) by RVSUM_1:17
      .=((-g)*P).i by A8,A10,FUNCT_1:12;
    hence thesis;
  end;
  hence thesis by A8,FINSEQ_1:13;
end;
