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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">U2</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</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>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</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>set </b><font color="Maroon">u2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font>;<br/>
<b>assume </b><a NAME="E1:39"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="alg_1.html#R3" target="_self">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 ;<br/>

<a NAME="E2:39"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:39"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a>  <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">U2</font> &amp;  <a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font> <a href="alg_1.html#R2" target="_self">is_monomorphism</a>  <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">U2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:39"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:39"/><i><font color="Green">E46</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:39"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>
 ; <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="E2:39_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="alg_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_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="E6:39_1"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</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>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">u</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">y</font>;<br/>
<a NAME="E7:39_1"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E8:39_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <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#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="alg_1.html#E1:39"><i><font color="Green">E33</font></i></a>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:39_1"/>
<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#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:39"/>
 <a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font> <a href="alg_1.html#R4" target="_self">is_isomorphism</a>  <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">U2</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E2:39"><i><font color="Green">E34</font></i></a>, <a class="txt" href="alg_1.html#E3:39"><i><font color="Green">E35</font></i></a>, </i>;<br/>


</div>
