reserve n,m for Nat;

theorem Th25:
  for f,g being real-valued FinSequence st
  f,g are_fiberwise_equipotent holds -f,-g are_fiberwise_equipotent
proof
  let f,g be real-valued FinSequence;
  assume
A1: f,g are_fiberwise_equipotent;
  then consider P being Permutation of dom g such that
A2: f = g*P by RFINSEQ:4;
A3: now
    per cases;
    case
      len g>=1;
      hence -f=(-g)*P by A2,Th24;
    end;
    case
      len g<1;
      then len g<0+1;
      then
A4:   len g = 0 by NAT_1:13;
      then
A5:   g={};
A6:   len f=0 by A1,A4,RFINSEQ:3;
      then len (-f)=0 by RVSUM_1:114;
      then
A7:   -f ={};
      f={} by A6;
      hence -f=(-g)*P by A2,A5,A7;
    end;
  end;
  dom (-g)=dom g by VALUED_1:8;
  hence thesis by A3,RFINSEQ:4;
end;
