<?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> , <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">A1</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">A2</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:44"/><i><font color="Green">E37</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font>
 ;<br/>

<b>let </b><font color="Maroon">h</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">A1</font>,<font color="Maroon">A2</font>;<br/>

<b>assume </b><a NAME="E2:44"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">h</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">A1</font>,<font color="Maroon">A2</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E3:44"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S'</font>,<font color="Maroon">S</font>
 ;<br/>

<b>set </b><font color="Maroon">B1</font> = <font color="Maroon">A1</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">B2</font> = <font color="Maroon">A2</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>;<br/>
<b>reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E5:44"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S'</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S'</font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <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/>
<b>take </b>
<font color="Maroon">h'</font>
;<br/>

<b>thus </b><a NAME="E7:44"/>
<font color="Maroon">h'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">h</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>
 ;<br/>

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a><br/>

<b>assume </b><a NAME="E8:44"/><i><font color="Green">E53</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E9:44"/><i><font color="Green">E56</font></i>: 
(  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">A1</font> &amp;  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">A2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:44"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , </i>;<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>,<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>;<br/>

<b>set </b><font color="Maroon">go</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">A1</font> <b>by </b><i>, </i>;<br/>
<a NAME="E12:44"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">h'</font> <a href="msualg_3.html#K5">#</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="msualg_3.html#K5">#</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E13:44"/><i><font color="Green">E102</font></i>: 
(  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">A1</font> &amp;  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">A2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:44"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S'</font> <a href="funct_2.html#R2">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
<font color="Maroon">h'</font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span> = 
<font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</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>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>
<b>hence </b><a NAME="E16:44"/>
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="msualg_3.html#K1">.</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>,<span class="p3">(<span class="default"><font color="Maroon">A1</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</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>,<span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="msualg_3.html#K5">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>;<br/>


</div>
