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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a> <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:88"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 ;<br/>

<a NAME="E2:88"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="group_6.html#V1">multiplicative</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="endalg.html#V1">unity-preserving</a> )
 
<b>by </b><i><a class="ref" href="quofield.html#D21" target="_self">Def21</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:88"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<i><font color="Green">E73</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E2:88"><i><font color="Green">E71</font></i></a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E3:88"><i><font color="Green">E72</font></i></a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="quofield.html#E2:88"><i><font color="Green">E71</font></i></a>, <a class="ref" href="endalg.html#D8">ENDALG:def 8</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:88_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:88_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/><b>hence </b><a NAME="E3:88_2"/>
contradiction
 <b>by </b><i><a class="txt" href="quofield.html#E4:88"><i><font color="Green">E73</font></i></a>, <a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:88"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="vectsp_1.html#K4">"</a> 
 <b>by </b><i><a class="txt" href="quofield.html#E4:88"><i><font color="Green">E73</font></i></a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>;<br/>


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