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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font> is <a href="relat_1.html#V2">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">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:43_1"/><i><font color="Green">E29</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="msualg_2.html#E1:43"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><a NAME="E4:43_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K8" target="_self">SubSort</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:43_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K8" target="_self">SubSort</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:43_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K5" target="_self">SubSort</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="msualg_2.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E4:43_1_1"/>
(  <a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="msualg_2.html#E3:43_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="msualg_2.html#T14" target="_self">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" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T18">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" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msualg_2.html#E3:43_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:43_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K8" target="_self">SubSort</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:43_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="msualg_2.html#E3:43_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D1">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" target="_self">MSSubSort</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="msualg_2.html#E5:43_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msualg_2.html#E7:43_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="msualg_2.html#D15" target="_self">Def15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:43"/>
 <a href="msualg_2.html#K9" target="_self">MSSubSort</a> <font color="Maroon">c<sub>3</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>


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