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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">U0</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E28">A1</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon" title="c4">i</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 <font color="Maroon" title="c1">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E3:43_1"/><i><font color="Green" title="E29">A2</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="msualg_2.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/><a NAME="E4:43_1"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">s</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:43_1_1"/>
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">B</font> being   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font><b> such that </b><br/><a NAME="E3:43_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K5" title="MSUALG_2:func.5">SubSort</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> )
 <b>by </b><i><a class="ref" href="msualg_2.html#D14" target="_self" title="MSUALG_2:def.14">Def14</a></i>;<br/>
<a NAME="E4:43_1_1"/>
(  <a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">B</font> )
 
<b>by </b><i><a class="txt" href="msualg_2.html#E3:43_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="msualg_2.html#T14" target="_self" title="MSUALG_2:th.14">Th14</a></i>;<br/>
<a NAME="E5:43_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></i>;<br/>
<b>hence </b><a NAME="E6:43_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="msualg_2.html#E3:43_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:43_1"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K8" title="MSUALG_2:func.8">SubSort</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:43_1"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="txt" href="msualg_2.html#E3:43_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>thus </b><a NAME="E8:43_1"/>
not <span class="p1">(<span class="default"><a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="msualg_2.html#E5:43_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="msualg_2.html#E7:43_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="msualg_2.html#D15" target="_self" title="MSUALG_2:def.15">Def15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:43"/>
 <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>by </b><i><a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>


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