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<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font>;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>

<a NAME="E2:81"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E1:81"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="quofield.html#D23" target="_self" title="QUOFIELD:def.23">Def23</a></i>;<br/>
<a NAME="E3:81"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E1:81"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="quofield.html#D23" target="_self" title="QUOFIELD:def.23">Def23</a></i>;<br/>
<a NAME="E4:81"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1"/><i><font color="Green" title="E62">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<a NAME="E2:81_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="quofield.html#E2:81"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="quofield.html#T53" target="_self" title="QUOFIELD:th.53">Th53</a></i>;<br/>
<b>hence </b><a NAME="E3:81_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="quofield.html#E3:81"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="quofield.html#E1:81_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:81"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="quofield.html#E2:81"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="quofield.html#T53" target="_self" title="QUOFIELD:th.53">Th53</a></i>;<br/>


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