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<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</a><br/> for <font color="Olive" title="b2">E</font> being   <a href="algspec1.html#M2" title="ALGSPEC1:mode.2">Extension</a> of <font color="Olive" title="b1">S</font>  st <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Olive" title="b2">E</font> holds <br/><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Olive" title="b1">S</font></span><br/><b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">E</font> being   <a href="algspec1.html#M2" title="ALGSPEC1:mode.2">Extension</a> of <font color="Maroon" title="c3">S</font>  st <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c3">S</font></span><br/><b>let </b><font color="Maroon" title="c4">E</font> be    <a href="algspec1.html#M2" title="ALGSPEC1:mode.2">Extension</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c4">E</font> implies <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c3">S</font> )</span><br/>







<b>set </b><font color="Maroon" title="c5">f'</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">E</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c1">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">g'</font> = the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">E</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">g</font>;<br/>
<b>set </b><font color="Maroon" title="c7">T</font> = <font color="Maroon" title="c4">E</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>;<br/>
<b>assume </b><a NAME="E1:60"/>
<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c3">S</font></span><br/>

<a NAME="E2:60"/><b>then </b><i><font color="Green" title="E47">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">E</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c1">f</font>,the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">E</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c4">E</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c3">S</font> is    <a href="instalg1.html#M1" title="INSTALG1:mode.1">Subsignature</a> of <font color="Maroon" title="c4">E</font> )
 
<b>by </b><i><a class="ref" href="algspec1.html#D5" target="_self" title="ALGSPEC1:def.5">Def5</a>, <a class="ref" href="algspec1.html#T41" target="_self" title="ALGSPEC1:th.41">Th41</a></i>;<br/>
<a NAME="E3:60"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">E</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">E</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c3">S</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T32" target="_self" title="ALGSPEC1:th.32">Th32</a>, <a class="ref" href="instalg1.html#T19" title="INSTALG1:th.19">INSTALG1:19</a></i>;<br/>
<a NAME="E4:60"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">E</font>
 
<b>by </b><i><a class="txt" href="algspec1.html#E2:60"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="instalg1.html#T11" title="INSTALG1:th.11">INSTALG1:11</a></i>;<br/>
<a NAME="E5:60"/><b>then </b><i><font color="Green" title="E49">A3</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">E</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T18" target="_self" title="ALGSPEC1:th.18">Th18</a></i>;<br/>
<a NAME="E6:60"/>
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">E</font>
 
<b>by </b><i><a class="txt" href="algspec1.html#E2:60"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="instalg1.html#T11" title="INSTALG1:th.11">INSTALG1:11</a></i>;<br/>
<a NAME="E7:60"/><b>then </b>
<span class="p1">(<span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">E</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c3">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T18" target="_self" title="ALGSPEC1:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E8:60"/>
<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="algspec1.html#E3:60"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="algspec1.html#E5:60"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="algspec1.html#T31" target="_self" title="ALGSPEC1:th.31">Th31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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