reserve n,m for Nat;

theorem
  for f1,f2 being real-valued FinSequence st len f1=len f2 & len f1>0 holds
  min (f1+f2)>=(min f1) +(min f2)
proof
  let f1,f2 be real-valued FinSequence;
  assume that
A1: len f1=len f2 and
A2: len f1>0;
A3: len (f1+f2)=len f1 by A1,RVSUM_1:115;
  then
A4: min_p (f1+f2) in dom (f1+f2) by A2,Def2;
  then 1<=min_p (f1+f2) & min_p (f1+f2)<=len (f1+f2) by FINSEQ_3:25;
  then
A5: min_p (f1+f2) in Seg len f1 by A3,FINSEQ_1:1;
  then min_p (f1+f2) in dom f2 by A1,FINSEQ_1:def 3;
  then
A6: f2.(min_p (f1+f2))>=f2.(min_p f2) by A1,A2,Def2;
  min_p (f1+f2) in dom f1 by A5,FINSEQ_1:def 3;
  then
A7: f1.(min_p (f1+f2))>=f1.(min_p f1) by A2,Def2;
  min (f1+f2)=f1.(min_p (f1+f2)) + f2.(min_p (f1+f2)) by A4,VALUED_1:def 1;
  hence thesis by A7,A6,XREAL_1:7;
end;
