reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem
  for f being increasing real-valued FinSequence holds max_p(f) = len f
  proof
    let f be increasing real-valued FinSequence;
    per cases;
    suppose len f = 0;
      hence thesis by RFINSEQ2:def 1;
    end;
    suppose
A1:   len f > 0;
      then len f >= 0+1 by NAT_1:13;
      then
A2:   len f in dom f by FINSEQ_3:25;
A3:   for i being Nat st i in dom f holds f.i <= f.len f
      proof
        let i be Nat such that
A4:     i in dom f;
        i <= len f by A4,FINSEQ_3:25;
        then per cases by XXREAL_0:1;
        suppose i = len f;
          hence thesis;
        end;
        suppose i < len f;
          hence thesis by A2,A4,VALUED_0:def 13;
        end;
      end;
      for j being Nat st j in dom f & f.j = f.len f holds len f <= j
      by A2,FUNCT_1:def 4;
      hence thesis by A1,A2,A3,RFINSEQ2:def 1;
    end;
  end;
