∀ x y : Z, [x + y] = [x - y] → y = 0
‘[=…]’ is an injection pattern
x, y: Z
Heq: x + y = x - yy = 0x, y: Z
Heq: y = - yy = 0
3 cases: y=0, y<0, y>0
x: Z
Heq: 0 = 00 = 0x: Z
p: positive
Heq: Z.pos p = Z.neg pZ.pos p = 0x: Z
p: positive
Heq: Z.neg p = Z.pos pZ.neg p = 0
y=0: trivial
x: Z
p: positive
Heq: Z.pos p = Z.neg pZ.pos p = 0x: Z
p: positive
Heq: Z.neg p = Z.pos pZ.neg p = 0
y!=0: contradiction in Heq
all: discriminate. Qed.