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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>, <font color="Maroon">S</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">R</font>,<font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:87"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">f</font> is <a href="quofield.html#V3" target="_self">RingMonomorphism</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<a NAME="E2:87"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 
<b>by </b><i><a class="ref" href="quofield.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E3:87"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="quofield.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E4:87"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:87_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="quofield.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E3:87_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:87"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> iff <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="quofield.html#D7" target="_self">Def7</a></i>;<br/>


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