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<b>let </b><font color="Maroon">UA</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Olive">x</font> ) )
 ;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="autalg_1.html#K3" target="_self">UAAutGroup</a> <font color="Maroon">UA</font>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="autalg_1.html#K6" target="_self">MSAAutGroup</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">UA</font></span>)</span>;<br/>
<a NAME="E2:49"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <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="autalg_1.html#K6" target="_self">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">UA</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K3" target="_self">UAAutGroup</a> <font color="Maroon">UA</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K6" target="_self">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">UA</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K3" target="_self">UAAutGroup</a> <font color="Maroon">UA</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K3" target="_self">UAAutGroup</a> <font color="Maroon">UA</font></span>)</span>;<br/><b>thus </b><a NAME="E1:49_1"/>
<font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font> ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">a'</font>, <font color="Maroon">B</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b'</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">UA</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font>, <font color="Maroon">B'</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K6" target="_self">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">UA</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ha</font> = <font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>, <font color="Maroon">hb</font> = <font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> as    <a href="autalg_1.html#M2" target="_self">Element</a> of  <a href="autalg_1.html#K4" target="_self">MSAAut</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">UA</font></span>)</span> ;<br/>
<a NAME="E5:49_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">ha</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">hb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 
<b>by </b><i><a class="txt" href="autalg_1.html#E5:49_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E6:49_1_1"/>
<font color="Maroon">b'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a'</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:49_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E5:49_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> = 
<font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="msuhom_1.html#K3">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E7:49_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msuhom_1.html#D3">MSUHOM_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3">MSAlg</a> <font color="Maroon">b'</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3">MSAlg</a> <font color="Maroon">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="msuhom_1.html#T26">MSUHOM_1:26</a></i>
<br/>.= 
<font color="Maroon">B</font> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3">MSAlg</a> <font color="Maroon">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="msuhom_1.html#D3">MSUHOM_1:def 3</a></i>
<br/>.= 
<font color="Maroon">B</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="msuhom_1.html#D3">MSUHOM_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:49"/><b>then </b>
<font color="Maroon">h'</font> is   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K3" target="_self">UAAutGroup</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K6" target="_self">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">UA</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:49"/>
<font color="Maroon">h</font> is   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K3" target="_self">UAAutGroup</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K6" target="_self">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">UA</font></span>)</span></span>)</span>
 ;<br/>


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