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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#NM2">Universal_Algebra</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="E1:22"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:22"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="alg_1.html#R3" target="_self">is_epimorphism</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies  <a href="alg_1.html#K1" target="_self">Image</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="alg_1.html#R3" target="_self">is_epimorphism</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then </b><i><font color="Green">E19</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> = 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="alg_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K1" target="_self">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="alg_1.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="alg_1.html#D6" target="_self">Def6</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K1" target="_self">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E4:22_1"/>
the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K1" target="_self">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1"/>
 <a href="alg_1.html#K1" target="_self">Image</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="alg_1.html#E2:22_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="alg_1.html#T2" target="_self">Th2</a></i>;<br/>


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

<b>assume </b><a NAME="E3:22"/>
 <a href="alg_1.html#K1" target="_self">Image</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then </b>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="alg_1.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="alg_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
;<br/>
<b>hence </b><a NAME="E5:22"/>
<font color="Maroon">c<sub>3</sub></font> <a href="alg_1.html#R3" target="_self">is_epimorphism</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="alg_1.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="alg_1.html#D3" target="_self">Def3</a></i>;<br/>


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