∀ 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.