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<b>let </b><font color="Maroon">U1</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">E</font> be    <a href="alg_1.html#M1" target="_self">Congruence</a> of <font color="Maroon">U1</font>;<br/>



<b>set </b><font color="Maroon">f</font> =  <a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font>;<br/>
<b>set </b><font color="Maroon">qa</font> =  <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font>;<br/>
<b>set </b><font color="Maroon">cqa</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">u1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>;<br/>
<b>thus </b><a NAME="E1:35"/>
 <a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>, <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="alg_1.html#D3" target="_self">ALG_1:def 3</a><br/>

<b>thus </b><a NAME="E2:35"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</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="E3:35"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:35"/><i><font color="Green">E34</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">U1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E</font>,<font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self">Th1</a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <b>by </b><i/>;<br/>
<a NAME="E8:35"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:35"/><b>then </b>
( <span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E</font>,<font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:35"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span>
 <b>by </b><i/>;<br/>


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