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<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">A</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">B</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <span class="p1"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> holds <br/><font color="Olive" title="b2">A</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a></span><br/><b>let </b><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">B</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <span class="p1"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> implies <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )</span><br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E4">A1</font></i>: 
( <font color="Maroon" title="c2">B</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <span class="p1"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a></span><br/>

<b>let </b><font color="Maroon" title="c4">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>assume </b><a NAME="E2:31"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<a NAME="E3:31"/><b>then </b>
<span class="p1"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E4">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="E4:31"/><b>then </b>
( <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="#E2:31"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="pboole.html#D21" title="PBOOLE:def.21">PBOOLE:def 21</a></i>;<br/>
<b>hence </b><a NAME="E5:31"/>
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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