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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">U1</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">U2</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">U3</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">U2</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U3</font>;<br/>









<b>assume </b><a NAME="E1:43"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">F</font> <a href="pboole.html#R6">=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="msualg_3.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U3</font> )
 ;<br/>

<a NAME="E2:43"/>
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>  st  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o</font>,<font color="Maroon">U1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o</font>,<font color="Maroon">U1</font> holds  <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:43_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font>;<br/>

<a NAME="E2:43_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E3:43_1_1"/><i><font color="Green">E34</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="msualg_3.html#E3:43_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S3</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> as  <a href="relat_1.html#V2">V2</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U2</font> <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E5:43_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E6:43_1_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E7:43_1_1"/><b>then </b>
<font color="Maroon">S3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E3:43_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E8:43_1_1"/><b>then </b>
<font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E4:43_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>hence </b><a NAME="E9:43_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:43_1"/><i><font color="Green">E35</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:43_1"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">a</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:43_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="msualg_3.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="msualg_3.html#E4:43_1"><i><font color="Green">E35</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E4:43_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E7:43_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">G</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E8:43_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U3</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">S3</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> as  <a href="relat_1.html#V2">V2</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U2</font> <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E10:43_1"/><i><font color="Green">E55</font></i>: 
( the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U3</font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">S3</font> &amp; <font color="Maroon">S3</font> is <a href="msualg_2.html#V3">opers_closed</a> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E11:43_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">S3</font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<i><font color="Green">E64</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U3</font> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">S3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T1" target="_self">Th1</a>, <a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<font color="Maroon">o</font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">S3</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U2</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S3</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#D3" target="_self">Def3</a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
;<br/>
<a NAME="E13:43_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S3</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U3</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/>
<a NAME="E14:43_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<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="E15:43_1"/><b>then </b>
<font color="Maroon">F</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U3</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:43_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="msualg_3.html#D2" target="_self">Def2</a>, <a class="ref" href="msualg_3.html#T2" target="_self">Th2</a>, , <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:43"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 <b>by </b><i/>;<br/>


</div>
