from __future__ import print_function
import io
line_1=open(r"my.txt")
line_2=open(r"not_my.txt")
word = u'>>'
with io.open('line_1', encoding='utf-8') as file:
    for line in file:
        if word in line:
            print(line, end='')

  
>>> lst = ['text1',
...        '>> text2',
...        'text3',
...        '>> text4']
>>> 
>>> for s in lst:
...     if s.startswith('>>'):
...         print(s[3:])
... 
text2
text4
>>>