reserve n,m for Nat;

theorem
  for f being real-valued FinSequence st len f>0 holds
  min (-f)=-(max f) & min_p (-f)=max_p f
proof
  let f be real-valued FinSequence;
  assume
A1: len f>0;
A2: len (-f)=len f by RVSUM_1:114;
  then
A3: min_p (-f) in dom (-f) by A1,Def2;
  then 1<=min_p (-f) & min_p (-f)<=len (-f) by FINSEQ_3:25;
  then min_p (-f) in Seg len f by A2,FINSEQ_1:1;
  then
A4: min_p (-f) in dom f by FINSEQ_1:def 3;
  then f.(max_p (f))>= f.(min_p (-f)) by A1,Def1;
  then
A5: -(f.(max_p (f)))<=-(f.(min_p (-f))) by XREAL_1:24;
A6: -(f.(max_p (f)))=(-f).(max_p (f)) & -(f.(min_p (-f)))=(-f).(min_p (-f))
  by RVSUM_1:17;
A7: dom (-f)=dom f by VALUED_1:8;
  then
A8: max_p (f) in dom (-f) by A1,Def1;
  then (-f).(min_p (-f))<=(-f).(max_p f) by A1,A2,Def2;
  then
A9: f.(min_p (-f))>=f.(max_p f) by A6,XREAL_1:24;
  f.(max_p (f))>=f.(min_p (-f)) by A1,A4,Def1;
  then f.(max_p (f))=f.(min_p (-f)) by A9,XXREAL_0:1;
  then
A10: min (-f)=-(f.(min_p (-f))) & min_p (-f)>=max_p f by A1,A7,A3,Def1,
RVSUM_1:17;
  max_p (f) in dom (-f) by A1,A7,Def1;
  then (-f).(min_p (-f))<=(-f).(max_p f) by A1,A2,Def2;
  then (-f).(min_p (-f))=(-f).(max_p f) by A6,A5,XXREAL_0:1;
  then min_p (-f)<=max_p f by A1,A2,A8,Def2;
  hence thesis by A10,XXREAL_0:1;
end;
