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<b>let </b><font color="Maroon">G</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L1">multLoopStr</a> ;<br/>

<b>reconsider </b><font color="Maroon">i</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> as   <a href="endalg.html#NM1">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E2:33_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:33_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:33_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:33_1_1_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E1:33_1_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:33_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E3:33_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="endalg.html#R1" target="_self">is_monomorphism</a>  &amp; <font color="Maroon">i</font> <a href="endalg.html#R2" target="_self">is_epimorphism</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:33_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="endalg.html#R3" target="_self">is_isomorphism</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:33_1_1"/>
 ex <font color="Olive">h</font> being  <a href="endalg.html#NM1">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font> st <font color="Olive">h</font> <a href="endalg.html#R3" target="_self">is_isomorphism</a> 
 ;<br/>


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