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<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">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>set </b><font color="Maroon">ff</font> = 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>;<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">S</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="E1:42"/>
<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/>

<a NAME="E2:42"/><b>then </b><i><font color="Green">E25</font></i>: 
( 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="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S</font>,<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> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</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> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</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> )
 
<b>by </b><i/>;<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 <span class="p1">(<span class="default"><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></span>)</span>;<br/>

<b>assume </b><a NAME="E3:42"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f'</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>
 ;<br/>

<b>let </b><font color="Maroon">g'</font> be    <a href="algspec1.html#M1" target="_self">rng-retract</a> of 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>thus </b>the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><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></span>)</span> = 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<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></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="algspec1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <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">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g'</font></span>)</span>
<b>by </b><i><a class="ref" href="algspec1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><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></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<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></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g'</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="lang1.html#K16">*</a> </span>)</span> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g'</font>
<b>by </b><i><a class="ref" href="algspec1.html#T2" target="_self">Th2</a>, <a class="ref" href="algspec1.html#T3" target="_self">Th3</a>, </i>
;<br/>


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