reserve n,m for Nat;

theorem
  for f being real-valued FinSequence, a being Real st len f>0 & a>0 holds
  min (a*f)=a*(min f) & min_p (a*f)=min_p f
proof
  let f be real-valued FinSequence, a be Real;
  assume that
A1: len f>0 and
A2: a>0;
A3: len (a*f)=len f by RVSUM_1:117;
  then
A4: min_p (a*f) in dom (a*f) by A1,Def2;
  then 1<=min_p (a*f) & min_p (a*f)<=len (a*f) by FINSEQ_3:25;
  then
A5: min_p (a*f) in dom f by A3,FINSEQ_3:25;
  then f.(min_p (f))<= f.(min_p (a*f)) by A1,Def2;
  then
A6: a*(f.(min_p (f)))<=a*(f.(min_p (a*f))) by A2,XREAL_1:64;
A7: a*(f.(min_p (f)))=(a*f).(min_p (f)) & a*(f.(min_p (a*f)))=(a*f).(min_p (
  a*f) ) by RVSUM_1:44;
A8: dom (a*f)=dom f by VALUED_1:def 5;
  then
A9: min_p (f) in dom (a*f) by A1,Def2;
  then (a*f).(min_p (f))>=(a*f).(min_p (a*f)) by A1,A3,Def2;
  then
A10: f.(min_p (f))>=f.(min_p (a*f)) by A2,A7,XREAL_1:68;
  f.(min_p (a*f))>=f.(min_p f) by A1,A5,Def2;
  then f.(min_p (f))=f.(min_p (a*f)) by A10,XXREAL_0:1;
  then
A11: min (a*f)=a*(f.(min_p (a*f))) & min_p (a*f)>=min_p f by A1,A8,A4,Def2,
RVSUM_1:44;
  min_p (f) in dom (a*f) by A1,A8,Def2;
  then (a*f).(min_p (a*f))<=(a*f).(min_p f) by A1,A3,Def2;
  then (a*f).(min_p (a*f))=(a*f).(min_p f) by A7,A6,XXREAL_0:1;
  then min_p (a*f)<=min_p f by A1,A3,A9,Def2;
  hence thesis by A11,XXREAL_0:1;
end;
