reserve n,m for Nat;

theorem
  for f1,f2 being real-valued FinSequence st len f1=len f2 & len f1>0 holds
  max (f1+f2)<=(max f1) +(max 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: max_p (f1+f2) in dom (f1+f2) by A2,Def1;
  then 1<=max_p (f1+f2) & max_p (f1+f2)<=len (f1+f2) by FINSEQ_3:25;
  then
A5: max_p (f1+f2) in Seg len f1 by A3,FINSEQ_1:1;
  then max_p (f1+f2) in dom f2 by A1,FINSEQ_1:def 3;
  then
A6: f2.(max_p (f1+f2))<=f2.(max_p f2) by A1,A2,Def1;
  max_p (f1+f2) in dom f1 by A5,FINSEQ_1:def 3;
  then
A7: f1.(max_p (f1+f2))<=f1.(max_p f1) by A2,Def1;
  max (f1+f2)=f1.(max_p (f1+f2)) + f2.(max_p (f1+f2)) by A4,VALUED_1:def 1;
  hence thesis by A7,A6,XREAL_1:7;
end;
