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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">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="Olive" title="b2">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> is    <a href="algspec1.html#M2" title="ALGSPEC1:mode.2">Extension</a> of <font color="Olive" title="b1">S</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></span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be  non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="Olive" title="b1">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> is    <a href="algspec1.html#M2" title="ALGSPEC1:mode.2">Extension</a> of <font color="Maroon" title="c3">S</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></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="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> is    <a href="algspec1.html#M2" title="ALGSPEC1:mode.2">Extension</a> of <font color="Maroon" title="c3">S</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> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">f9</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/>
<span class="kw">set </span><font color="Maroon" title="c6">g9</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="c2">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">gg</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="c2">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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/>
<a NAME="E1:62"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="c2">g</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="c2">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#T11" target="_self" title="ALGSPEC1:th.11">Th11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:62"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> is    <a href="algspec1.html#M2" title="ALGSPEC1:mode.2">Extension</a> of <font color="Maroon" title="c3">S</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></span><br/>

<a NAME="E3:62"/><span class="kw">then </span>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#T54" target="_self" title="ALGSPEC1:th.54">Th54</a></span>;<br/>
<a NAME="E4:62"/><span class="kw">then </span>
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>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="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 class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#T31" target="_self" title="ALGSPEC1:th.31">Th31</a></span>;<br/>
<a NAME="E5:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c1">f</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="algspec1.html#T31" target="_self" title="ALGSPEC1:th.31">Th31</a></span>;<br/>
<a NAME="E6:62"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#D5" target="_self" title="ALGSPEC1:def.5">Def5</a></span>;<br/>
<a NAME="E7:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="c2">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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#U4" title="STRUCT_0:sel.4">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="c2">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#T18" target="_self" title="ALGSPEC1:th.18">Th18</a>, <a class="ref" href="instalg1.html#T11" title="INSTALG1:th.11">INSTALG1:11</a></span>;<br/>
<a NAME="E8:62"/>
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="struct_0.html#U4" title="STRUCT_0:sel.4">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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="algspec1.html#T41" target="_self" title="ALGSPEC1:th.41">Th41</a></span>;<br/>
<a NAME="E9:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c3">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <span class="p1">(<span class="default"><span class="p2">(<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>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="c2">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">S</font></span>)</span> is    <a href="instalg1.html#M1" title="INSTALG1:mode.1">Subsignature</a> of <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="algspec1.html#T40" target="_self" title="ALGSPEC1:th.40">Th40</a>, <a class="ref" href="instalg1.html#T19" title="INSTALG1:th.19">INSTALG1:19</a></span>;<br/>
<a NAME="E10:62"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="algspec1.html#T18" target="_self" title="ALGSPEC1:th.18">Th18</a>, <a class="ref" href="instalg1.html#T11" title="INSTALG1:th.11">INSTALG1:11</a></span>;<br/>
<a NAME="E11:62"/><span class="kw">then </span>
<font color="Maroon" title="c3">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <span class="p1">(<span class="default"><span class="p2">(<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>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="c2">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="algspec1.html#T44" target="_self" title="ALGSPEC1:th.44">Th44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:62"/>
<font color="Maroon" title="c3">S</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> is    <a href="instalg1.html#M1" title="INSTALG1:mode.1">Subsignature</a> of <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="algspec1.html#T45" target="_self" title="ALGSPEC1:th.45">Th45</a>, <a class="ref" href="algspec1.html#T54" target="_self" title="ALGSPEC1:th.54">Th54</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="algspec1.html#D5" target="_self" title="ALGSPEC1:def.5">ALGSPEC1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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