Is this approach right in proving a theorem
Ques: show that a mod m = b mod m if a is congruent to b (mod m)
Proof: given a is congruent to b(mod m)
According to definition:
a - b / m
i.e a - b = mx (for some integer x). ....(1)
Also a = mx + b
that can also be written as a mod m = b ...(2)
Also again from equation (1)
-b = mx - a
b = -mx + a
Or b = m(-x) + a
Or b = my + a ...(3)
Therefore b mod m = a
If we replace b in equation (2) by equation (3) then we get
a mod m = my + a
my + a can also be written as b mod m
Therefore, a mod m = b mod m