<?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">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="algspec1.html#R1" target="_self">form_a_replacement_in</a> <font color="Maroon">S</font>
 ;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">o2</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><a NAME="E2:32_1"/><i><font color="Green">E26</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 ;<br/><b>assume </b><a NAME="E3:32_1"/>
<span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font>
 ;<br/><b>then </b><i><font color="Green">E27</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font>
<b>by </b><i/>
;<br/><a NAME="E5:32_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
 <b>by </b><i><a class="txt" href="algspec1.html#E4:32_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="algspec1.html#E1:32_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E2:32"><i><font color="Green">E28</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E2:32"><i><font color="Green">E28</font></i></a>, </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E4:32_1"><i><font color="Green">E27</font></i></a>, , <a class="txt" href="algspec1.html#E1:32_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i/>
;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:32"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>  st <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">o1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">o2</font> holds <br/>( <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o2</font></span>)</span> &amp; <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o2</font></span>)</span> )
 ;<br/>

<b>let </b><font color="Maroon">o1</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">o2</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="algspec1.html#D3" target="_self">ALGSPEC1:def 3</a><br/>

<a NAME="E3:32"/><i><font color="Green">E30</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 
;<br/>
<b>assume </b><a NAME="E4:32"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font>
 ;<br/>

<b>then </b><i><font color="Green">E31</font></i>: <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font>
<b>by </b><i/>
;<br/>
<a NAME="E6:32"/><b>then </b>
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> = 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E2:32"><i><font color="Green">E28</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E2:32"><i><font color="Green">E28</font></i></a>, </i>
;<br/><br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span> = 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i/>
;<br/>


</div>
