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<b>let </b><font color="Maroon" title="c1">UA</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">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E32">A1</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font> holds <br/><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Olive" title="b1">x</font> ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="autalg_1.html#K3" title="AUTALG_1:func.3">UAAutGroup</a> <font color="Maroon" title="c1">UA</font>;<br/>
<b>set </b><font color="Maroon" title="c4">H</font> =  <a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span>;<br/>
<a NAME="E2:49"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">h</font> <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="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:49"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="autalg_1.html#E52"><i><font color="Green" title="E30">Lm4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">h'</font> = <font color="Maroon" title="c2">h</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K3" title="AUTALG_1:func.3">UAAutGroup</a> <font color="Maroon" title="c1">UA</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="autalg_1.html#E1:49"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">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" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K3" title="AUTALG_1:func.3">UAAutGroup</a> <font color="Maroon" title="c1">UA</font></span>)</span>;<br/><b>thus </b><a NAME="E1:49_1"/>
<font color="Maroon" title="c5">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">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" title="c8">a'</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c7">b</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">A</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c11">B</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c9">b'</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span> <b>by </b><i><a class="ref" href="autalg_1.html#T39" target="_self" title="AUTALG_1:th.39">Th39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">A'</font> = <font color="Maroon" title="c10">A</font>, <font color="Maroon" title="c13">B'</font> = <font color="Maroon" title="c11">B</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="autalg_1.html#T34" target="_self" title="AUTALG_1:th.34">Th34</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">ha</font> = <font color="Maroon" title="c5">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c15">hb</font> = <font color="Maroon" title="c5">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font> as    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span> ;<br/>
<a NAME="E5:49_1_1"/><i><font color="Green" title="E33">A2</font></i>: 
( <font color="Maroon" title="c14">ha</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A'</font> &amp; <font color="Maroon" title="c15">hb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B'</font> )
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:49"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/>
<a NAME="E6:49_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c5">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">a'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">a'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:49"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="autalg_1.html#T7" target="_self" title="AUTALG_1:th.7">Th7</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font></span>)</span> = 
<font color="Maroon" title="c5">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></i>
<br/>.= 
 <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">a'</font></span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E6:49_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="msuhom_1.html#D3" title="MSUHOM_1:def.3">MSUHOM_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c9">b'</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c8">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="msuhom_1.html#T26" title="MSUHOM_1:th.26">MSUHOM_1:26</a></i>
<br/>.= 
<font color="Maroon" title="c11">B</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c8">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="msuhom_1.html#D3" title="MSUHOM_1:def.3">MSUHOM_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c11">B</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c10">A</font>
<b>by </b><i><a class="ref" href="msuhom_1.html#D3" title="MSUHOM_1:def.3">MSUHOM_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E5:49_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:49"/>
<font color="Maroon" title="c2">h</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K3" title="AUTALG_1:func.3">UAAutGroup</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/>


</div>
