reserve n,m for Nat;

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