<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> : <font color="Olive">x</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> </span>}</span> ;<br/>
<a NAME="E1:1_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> : <font color="Olive">x</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>set </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">UA</font>;<br/>
<a NAME="E1:1_1_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E2:1_1_1_1"/>
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#T6">ALG_1:6</a></i>;<br/>
<b>hence </b><a NAME="E3:1_1_1_1"/>
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> : <font color="Olive">x</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> : <font color="Olive">x</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> </span>}</span>  is <a href="fraenkel.html#V1">functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a><br/>

<b>assume </b><a NAME="E1:1_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> : <font color="Olive">x</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font><b> such that </b><br/><a NAME="E3:1_1_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> )
 ;<br/>
<b>thus </b><a NAME="E4:1_1_1_2"/>
<font color="Maroon">q</font> is    <a href="hidden.html#M1">set</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> : <font color="Olive">x</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<a NAME="E4:1_1_1"/>
<font color="Maroon">F</font> is    <a href="fraenkel.html#M1">FUNCTION_DOMAIN</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fraenkel.html#D2">FRAENKEL:def 2</a><br/>

<a NAME="E1:1_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font><b> such that </b><br/><a NAME="E3:1_1_1_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> )
 ;<br/>
<b>thus </b><a NAME="E4:1_1_1_3"/>
<font color="Maroon">a</font> is   <a href="relset_1.html#M2">M5</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>)
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="fraenkel.html#M1">FUNCTION_DOMAIN</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">UA</font>,<font color="Maroon">UA</font>;<br/>

<b>thus </b><a NAME="E6:1_1_1"/>
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies <font color="Maroon">h</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_4"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:1_1_1_4"/><b>then </b>
 ex <font color="Olive">f</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> st <br/>( <font color="Olive">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font> &amp; <font color="Olive">f</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font> )
 
;<br/>
<b>hence </b><a NAME="E3:1_1_1_4"/>
<font color="Maroon">h</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font>
 ;<br/>


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

<b>assume </b><a NAME="E7:1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">h</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font>
 ;<br/>

<a NAME="E8:1_1_1"/>
<font color="Maroon">h</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E9:1_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


</div>
