Given \(a=xy^{p-1},\; b=xy^{q-1},\; c=xy^{r-1}\), prove:
\[ a^{q-r} b^{r-p} c^{p-q} = 1 \]
Proof
\[ = (xy^{p-1})^{q-r} (xy^{q-1})^{r-p} (xy^{r-1})^{p-q} \]
\[ = x^{q-r}y^{(p-1)(q-r)} \cdot x^{r-p}y^{(q-1)(r-p)} \cdot x^{p-q}y^{(r-1)(p-q)} \]
\[ = x^{(q-r)+(r-p)+(p-q)} \cdot y^{(p-1)(q-r)+(q-1)(r-p)+(r-1)(p-q)} \]
\[ = x^0 \cdot y^0 \]
\[ = 1 \]