reserve a,b,c for set;

theorem Th4:
  for D being non empty set,f being FinSequence of D, n,m being
  Element of NAT st n<=m holds len (f/^m)<=len (f/^n)
proof
  let D be non empty set,f be FinSequence of D,n,m be Element of NAT;
A1: len (f/^m) = len f -' m by RFINSEQ:29;
  assume
A2: n<=m;
  then
A3: n-n<=m-n by XREAL_1:9;
  now
    per cases;
    suppose
A4:   len f <= n;
      then len f -' n - (len f -' m) = len f -' n - 0 by A2,NAT_2:8,XXREAL_0:2
        .= 0 by A4,NAT_2:8;
      hence len (f/^n) - len (f/^m) >= 0 by A1,RFINSEQ:29;
    end;
    suppose
A5:   len f > n;
      per cases;
      suppose
        len f <= m;
        then len f -' n - (len f -' m) = len f -' n - 0 by NAT_2:8
          .= len f -' n;
        hence len (f/^n) - len (f/^m) >= 0 by A1;
      end;
      suppose
        len f > m;
        then len f-'n-(len f-'m)= len f-'n-(len f-m) by XREAL_1:233
          .= len f-n-(len f-m) by A5,XREAL_1:233
          .= m-n;
        hence len (f/^n) - len (f/^m) >= 0 by A3,A1,RFINSEQ:29;
      end;
    end;
  end;
  then len (f/^n) - len (f/^m) + len (f/^m) >= 0 + len (f/^m) by XREAL_1:6;
  hence thesis;
end;
