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<b>consider </b><font color="Maroon" title="c3">G</font> being    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E2:43_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 <b>by </b><i><a class="ref" href="msafree2.html#D10" title="MSAFREE2:def.10">MSAFREE2:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">B</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E4:43_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="pboole.html#R3" title="PBOOLE:pred.3">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#T146" title="PBOOLE:th.146">PBOOLE:146</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c5">C</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E6:43_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> holds <br/><font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_1_1"/>
<font color="Maroon" title="c6">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/><a NAME="E2:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:43_1_1"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_1"/>
not <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E8:43_1_1"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c5">C</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/>
<b>set </b><font color="Maroon" title="c6">H</font> = <font color="Maroon" title="c3">G</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">C</font>;<br/>
<a NAME="E9:43_1_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c5">C</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">G</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">C</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></i>;<br/>
<a NAME="E10:43_1_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_1_2"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c7">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/><a NAME="E2:43_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E4:43_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a></i>;<br/><a NAME="E3:43_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E4:43_1_1_2"/>
<font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:43_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="msscyc_1.html#E1:43_1_1_2"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:43_1_1"/><b>then </b>
<font color="Maroon" title="c5">C</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E13:43_1_1"/><b>then </b>
<font color="Maroon" title="c3">G</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">C</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E10:43_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c3">G</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">C</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="msscyc_1.html#E8:43_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="msscyc_1.html#E9:43_1_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="pboole.html#T143" title="PBOOLE:th.143">PBOOLE:143</a>, <a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E15:43_1_1"/>
<font color="Maroon" title="c3">G</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">H</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">H</font> = <font color="Maroon" title="c7">H</font> as    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msscyc_1.html#T21" target="_self" title="MSSCYC_1:th.21">Th21</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">H</font>
;<br/>

<b>thus </b><a NAME="E17:43_1_1"/>
<font color="Maroon" title="c8">H</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a><br/>

<b>assume </b><a NAME="E18:43_1_1"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c9">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/>

<a NAME="E19:43_1_1"/><b>then </b><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E2:43_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a></i>;<br/>
<a NAME="E20:43_1_1"/><i><font color="Green" title="E29">A10</font></i>: 
<font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E6:43_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="msscyc_1.html#E18:43_1_1"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/>
<a NAME="E21:43_1_1"/>
<font color="Maroon" title="c8">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E18:43_1_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="pboole.html#D7" title="PBOOLE:def.7">PBOOLE:def 7</a></i>;<br/>
<b>hence </b><a NAME="E22:43_1_1"/>
<font color="Maroon" title="c8">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E19:43_1_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="msscyc_1.html#E20:43_1_1"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a></i>;<br/>


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