Correctness of Extended-Euclid

d' = gcd(b, a mod b)
= bx' + (a mod b)y' $\;\;\;\;\;$ ; Euclid's Theorem
= d = gcd(a, b) $\;\;\;\;\;$ ; Euclid's Algorithm

d = bx' + (a - $\lfloor$ $\frac{a}{b}$ $\rfloor$ b) y' $\;\;\;\;\;$ ; Division Theorem
= ay' + b(x' - $\lfloor$ $\frac{a}{b}$ $\rfloor$ y') $\;\;\;\;\;$ ; Rearrange terms

d = d' = ax + by
x = y'
y = x' - $\lfloor$ $\frac{a}{b}$ $\rfloor$ y'