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<span class="kw">set </span><font color="Maroon" title="c4">g1</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="c3">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">g2</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="c3">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f1</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="c2">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">f2</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="c2">f</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">S1</font>, <font color="Maroon" title="c9">S2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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="c2">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="c3">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="c8">S1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">S1</font> <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="c2">f</font></span>)</span> &amp; the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">S1</font> <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="c3">g</font></span>)</span> &amp; 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="c2">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="c3">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="c9">S2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S2</font> <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="c2">f</font></span>)</span> &amp; the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c9">S2</font> <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="c3">g</font></span>)</span> implies <font color="Maroon" title="c8">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38_1_1"/><span class="lab"><font color="Green" title="E29">A2</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="c2">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="c3">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="c8">S1</font>
 <span class="kw">and </span><br/><a NAME="E2:38_1_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">S1</font> <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="c2">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:38_1_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">S1</font> <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="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:38_1_1"/><span class="lab"><font color="Green" title="E32">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="c2">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="c3">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="c9">S2</font>
 <span class="kw">and </span><br/><a NAME="E5:38_1_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S2</font> <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="c2">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:38_1_1"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c9">S2</font> <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="c3">g</font></span>)</span>
 ; <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="c8">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S2</font></span><br/>

<a NAME="E7:38_1_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c8">S1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c9">S2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:38_1_1_1"/>
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">S1</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="c9">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E6:38_1_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_2.html#D8" title="FUNCT_2:def.8">FUNCT_2:def 8</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">S1</font> holds  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c8">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <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="c9">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c8">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c8">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> <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="c9">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">o1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:38_1_1_1"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c11">o1</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> <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="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:38_1_1_1"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:38_1_1_1"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c12">o2</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> <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="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:38_1_1_1"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">o1</font> = <font color="Maroon" title="c11">o1</font>, <font color="Maroon" title="c14">o2</font> = <font color="Maroon" title="c12">o2</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="#E1:38_1_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:38_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E6:38_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></span>;<br/>
<span class="kw">thus </span>the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c8">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">o</font> = 
<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="c8">S1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <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="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E4:38_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<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="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">f</font></span>)</span> <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="c13">o1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></span>
<br/>.= 
<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="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<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="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c14">o2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E4:38_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E7:38_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="algspec1.html#T30" target="_self" title="ALGSPEC1:th.30">Th30</a></span>
<br/>.= 
<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="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">f</font></span>)</span> <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="c14">o2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<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="c9">S2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <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="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">o2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></span>
<br/>.= 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c9">S2</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="#E6:38_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E7:38_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:38_1_1"/>
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c8">S1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c9">S2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:38_1_1_2"/>
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">S1</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="c9">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E6:38_1_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_2.html#D8" title="FUNCT_2:def.8">FUNCT_2:def 8</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">S1</font> holds  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c8">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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 <font color="Maroon" title="c9">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c8">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c8">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</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 <font color="Maroon" title="c9">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:38_1_1_2"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c11">o2</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> <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="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:38_1_1_2"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">o2</font> = <font color="Maroon" title="c11">o2</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="#E1:38_1_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:38_1_1_2"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></span>;<br/>
<span class="kw">thus </span>the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c8">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">o</font> = 
<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="c2">f</font></span>)</span> <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="c12">o2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E4:38_1_1_2"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></span>
<br/>.= 
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c9">S2</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="#E4:38_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E4:38_1_1_2"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:38_1_1"/>
<font color="Maroon" title="c8">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E5:38_1_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E6:38_1_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E7:38_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
