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<b>let </b><font color="Maroon" title="c1">U1</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">E</font> be    <a href="alg_1.html#M1" title="ALG_1:mode.1">Congruence</a> of <font color="Maroon" title="c1">U1</font>;<br/>



<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>;<br/>
<b>set </b><font color="Maroon" title="c4">qa</font> =  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>;<br/>
<b>set </b><font color="Maroon" title="c5">cqa</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">u1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>;<br/>
<b>thus </b><a NAME="E1:35"/>
 <a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>, <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>
 <b>by </b><i><a class="ref" href="alg_1.html#T21" target="_self" title="ALG_1:th.21">Th21</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="alg_1.html#D3" target="_self" title="ALG_1:def.3">ALG_1:def 3</a><br/>

<b>thus </b><a NAME="E2:35"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:35"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c7">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <b>by </b><i><a class="txt" href="alg_1.html#E3:35"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:35"/><i><font color="Green" title="E29">A2</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> &amp; <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="txt" href="alg_1.html#E3:35"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <b>by </b><i><a class="txt" href="alg_1.html#E6:35"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>
<a NAME="E8:35"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">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" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c10">y</font> )
 
<b>by </b><i><a class="ref" href="alg_1.html#D15" target="_self" title="ALG_1:def.15">Def15</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:35"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="alg_1.html#E6:35"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>


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