reserve n,m for Nat;

theorem
  for f being FinSequence of REAL,n being Nat st n<len f
  holds max (f/^n)<= max f & min (f/^n)>= min f
proof
  let f be FinSequence of REAL,n be Nat;
A1: 1<=1+n by NAT_1:12;
  assume
A2: n<len f;
  then
A3: len (f/^n)=len f -n by RFINSEQ:def 1;
  then
A4: len (f/^n)>0 by A2,XREAL_1:50;
  then
A5: (min_p (f/^n)) in dom (f/^n) by Def2;
  then
A6: f.( (min_p (f/^n))+n)=min (f/^n) by A2,RFINSEQ:def 1;
  (min_p (f/^n))<=len (f/^n) by A5,FINSEQ_3:25;
  then
A7: (min_p (f/^n))+n<=len f-n+n by A3,XREAL_1:7;
A8: 1<=1+n by NAT_1:12;
  1<= (min_p (f/^n)) by A5,FINSEQ_3:25;
  then 1+n<=(min_p (f/^n))+n by XREAL_1:7;
  then 1<= ((min_p (f/^n))+n) by A8,XXREAL_0:2;
  then
A9: ((min_p (f/^n))+n) in dom f by A7,FINSEQ_3:25;
A10: (max_p (f/^n)) in dom (f/^n) by A4,Def1;
  then (max_p (f/^n))<=len (f/^n) by FINSEQ_3:25;
  then
A11: (max_p (f/^n))+n<=len f-n+n by A3,XREAL_1:7;
  1<= (max_p (f/^n)) by A10,FINSEQ_3:25;
  then 1+n<=(max_p (f/^n))+n by XREAL_1:7;
  then 1<= ((max_p (f/^n))+n) by A1,XXREAL_0:2;
  then
A12: ((max_p (f/^n))+n) in dom f by A11,FINSEQ_3:25;
  f.( (max_p (f/^n))+n)=max (f/^n) by A2,A10,RFINSEQ:def 1;
  hence thesis by A2,A12,A9,A6,Def1,Def2;
end;
