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

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="msualg_1.html#V2">void</a> <a href="catalg_1.html#NM1">Signature</a>;<br/><b>let </b><font color="Maroon">E</font> be    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">S</font>;<br/>







<b>set </b><font color="Maroon">f'</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">g'</font> = the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">gg</font> = 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>;<br/>
<b>set </b><font color="Maroon">T</font> = <font color="Maroon">E</font> <a href="algspec1.html#K2" target="_self">with-replacement</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>;<br/>
<b>assume </b><a NAME="E1:62"/><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">E</font>
 ;<br/>

<a NAME="E2:62"/><b>then </b><i><font color="Green">E26</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>
 
<b>by </b><i><a class="ref" href="algspec1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E3:62"/><b>then </b>
( 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>,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> <a href="algspec1.html#R1" target="_self">form_a_replacement_in</a> <font color="Maroon">S</font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</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="pboole.html#R6">=</a> 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> )
 
<b>by </b><i>, <a class="txt" href="algspec1.html#E4:62"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E4:62"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font>,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> <a href="algspec1.html#R1" target="_self">form_a_replacement_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="algspec1.html#E4:62"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E5:62"/><i><font color="Green">E28</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">E</font>,<font color="Maroon">E</font> <a href="algspec1.html#K2" target="_self">with-replacement</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> &amp; <font color="Maroon">S</font> is    <a href="instalg1.html#M1">Subsignature</a> of <font color="Maroon">E</font> )
 
<b>by </b><i>, , <a class="ref" href="algspec1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:62"/><b>then </b>
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S</font>,<font color="Maroon">E</font> <a href="algspec1.html#K2" target="_self">with-replacement</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="instalg1.html#T19">INSTALG1:19</a></i>;<br/>
<a NAME="E7:62"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">S</font> <a href="algspec1.html#K2" target="_self">with-replacement</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span> is    <a href="instalg1.html#M1">Subsignature</a> of <font color="Maroon">E</font> <a href="algspec1.html#K2" target="_self">with-replacement</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:62"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T7" target="_self">Th7</a>, <a class="ref" href="instalg1.html#T11">INSTALG1:11</a></i>;<br/>
<a NAME="E9:62"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> 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>
 
<b>by </b><i/>;<br/>
<a NAME="E10:62"/>
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T7" target="_self">Th7</a>, <a class="ref" href="instalg1.html#T11">INSTALG1:11</a></i>;<br/>
<a NAME="E11:62"/><b>then </b>
<span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> 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>
 
<b>by </b><i/>;<br/>
<a NAME="E12:62"/><b>then </b>
<font color="Maroon">S</font> <a href="algspec1.html#K2" target="_self">with-replacement</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="algspec1.html#K2" target="_self">with-replacement</a> <font color="Maroon">f</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>
 
<b>by </b><i>, <a class="ref" href="algspec1.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E13:62"/>
<font color="Maroon">S</font> <a href="algspec1.html#K2" target="_self">with-replacement</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> is    <a href="instalg1.html#M1">Subsignature</a> of <font color="Maroon">E</font> <a href="algspec1.html#K2" target="_self">with-replacement</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="algspec1.html#T8" target="_self">Th8</a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="algspec1.html#D5" target="_self">ALGSPEC1:def 5</a><br/>


</div>
