Whakaoti i te A(AADB)/A(ACDB)=AO/CO

$$\frac { A ( A A D B ) } { A ( A C D B ) } = \frac { A O } { C O }$$

Whakaoti mō A

$$A\neq 0$$

$$C\neq 0\text{ and }D\neq 0\text{ and }B\neq 0\text{ and }O\neq 0$$

Ngā Upane Otinga

Tē taea kia ōrite te tāupe A ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te BCDOA^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o AACDB,CO.

$$OAAADB=BDA^{2}AO$$

Whakareatia te A ki te A, ka A^{2}.

$$OA^{2}ADB=BDA^{2}AO$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 2 me te 1 kia riro ai te 3.

$$OA^{3}DB=BDA^{2}AO$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 2 me te 1 kia riro ai te 3.

$$OA^{3}DB=BDA^{3}O$$

Tangohia te BDA^{3}O mai i ngā taha e rua.

$$OA^{3}DB-BDA^{3}O=0$$

Pahekotia te OA^{3}DB me -BDA^{3}O, ka 0.

$$0=0$$

Whakatauritea te 0 me te 0.

$$\text{true}$$

He pono tēnei mō tētahi A ahakoa.

$$A\in \mathrm{R}$$

Tē taea kia ōrite te tāupe A ki 0.

$$A\in \mathrm{R}\setminus 0$$

Whakaoti mō B

$$B\neq 0$$

$$C\neq 0\text{ and }D\neq 0\text{ and }O\neq 0\text{ and }A\neq 0$$

Ngā Upane Otinga

Tē taea kia ōrite te tāupe B ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te BCDOA^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o AACDB,CO.

$$OAAADB=BDA^{2}AO$$

Whakareatia te A ki te A, ka A^{2}.

$$OA^{2}ADB=BDA^{2}AO$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 2 me te 1 kia riro ai te 3.

$$OA^{3}DB=BDA^{2}AO$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 2 me te 1 kia riro ai te 3.

$$OA^{3}DB=BDA^{3}O$$

Tangohia te BDA^{3}O mai i ngā taha e rua.

$$OA^{3}DB-BDA^{3}O=0$$

Pahekotia te OA^{3}DB me -BDA^{3}O, ka 0.

$$0=0$$

Whakatauritea te 0 me te 0.

$$\text{true}$$

He pono tēnei mō tētahi B ahakoa.

$$B\in \mathrm{R}$$

Tē taea kia ōrite te tāupe B ki 0.

$$B\in \mathrm{R}\setminus 0$$