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<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">S9</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">S9</font> holds <br/><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>) is    <a href="instalg1.html#M1" title="INSTALG1:mode.1">Subsignature</a> of <font color="Maroon" title="c2">S9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">S9</font> implies <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>) is    <a href="instalg1.html#M1" title="INSTALG1:mode.1">Subsignature</a> of <font color="Maroon" title="c2">S9</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">R</font> = <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">F</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">G</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">f9</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">g9</font> =  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c4">g</font>;<br/>
<a NAME="E1:42"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E2:42"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</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="c2">S9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:42"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:42"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">S9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>) is    <a href="instalg1.html#M1" title="INSTALG1:mode.1">Subsignature</a> of <font color="Maroon" title="c2">S9</font></span><br/>

<a NAME="E5:42"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</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="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/>
<a NAME="E6:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A5</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#T10" target="_self" title="ALGSPEC1:th.10">Th10</a></span>;<br/>
<a NAME="E7:42"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="algspec1.html#T31" target="_self" title="ALGSPEC1:th.31">Th32</a></span>;<br/>
<a NAME="E8:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A7</font></span>: 
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#D4" target="_self" title="ALGSPEC1:def.4">Def4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:42"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a>,<a class="ref" href="instalg1.html#D2" title="INSTALG1:def.2">INSTALG1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <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="c2">S9</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S9</font> &amp;  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) )</span><br/>

<a NAME="E10:42"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E11:42"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</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="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/>
<a NAME="E12:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A9</font></span>: 
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#T10" target="_self" title="ALGSPEC1:th.10">Th10</a></span>;<br/>
<a NAME="E13:42"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">f</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c4">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="algspec1.html#D4" target="_self" title="ALGSPEC1:def.4">Def4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E44">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> implies  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E4:42_1"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c11">a</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/><a NAME="E6:42_1"/>
 the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/><a NAME="E7:42_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A15</font></span>: 
 the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><a NAME="E8:42_1"/>
 the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/><a NAME="E9:42_1"/><span class="kw">then </span>
 the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_1"/>
 the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="txt" href="#E7:42_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:42"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/>
<a NAME="E16:42"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E17:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A16</font></span>: 
 the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<a NAME="E18:42"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <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="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="algspec1.html#D4" target="_self" title="ALGSPEC1:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:42"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <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="c2">S9</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) )</span><br/>

<a NAME="E20:42"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 
;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> = 
 the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T53" title="RELAT_1:th.53">RELAT_1:53</a></span>
<br/>.= 
 the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> or <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">p</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> or  not <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> or <font color="Maroon" title="c11">p</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E22:42"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E23:42"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">p</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E25:42"/><span class="lab"><font color="Green" title="E48">A19</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E26:42"/><span class="lab"><font color="Green" title="E49">A20</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E22:42"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c12">a</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E25:42"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/>
<a NAME="E28:42"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E26:42"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/>
<a NAME="E29:42"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:42"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E23:42"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<a NAME="E30:42"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E31:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A22</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">p</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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E32:42"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E23:42"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E26:42"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">p</font> = 
 the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:42"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="ref" href="relat_1.html#T53" title="RELAT_1:th.53">RELAT_1:53</a></span>
<br/>.= 
 the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">S9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:42"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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