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<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="membered.html#V5" title="MEMBERED:attr.5">integer-membered</a>  <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"> ( <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> implies <font color="Maroon" title="c1">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_1_1"/>
( <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_1_1_1"/>
<font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> implies <font color="Maroon" title="c1">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/><div class="add"><b>hence </b><a NAME="E2:77_1_1_1_1"/>
( <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> implies <font color="Maroon" title="c1">X</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_1_1_2"/><i><font color="Green" title="E54">S</font></i>: 
not <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> implies <font color="Maroon" title="c1">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/><div class="add"><b>assume </b><a NAME="E2:77_1_1_1_2"/>
( <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xxreal_2.html#D11" target="_self" title="XXREAL_2:def.11">XXREAL_2:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/><b>then reconsider </b><font color="Maroon" title="c2">Z</font> = <font color="Maroon" title="c1">X</font> as  <a href="membered.html#V5" title="MEMBERED:attr.5">integer-membered</a> non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E1:77_1_1_1_2"><i><font color="Green" title="E54">S</font></i></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>  = <font color="Maroon">c<sub>1</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>;<br/><b>set </b><font color="Maroon" title="c3">m</font> =  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">Z</font>;<br/><b>set </b><font color="Maroon" title="c4">n</font> =  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c2">Z</font>;<br/><b>set </b><font color="Maroon" title="c5">Y</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  : (  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">Z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span>}</span> ;<br/><a NAME="E4:77_1_1_1_2"/><i><font color="Green" title="E55">A</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  : (  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">Z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="xxreal_2.html#S1" target="_self" title="XXREAL_2:sch.1">XXREAL_2:sch 1</a>();<br/></i><a NAME="E5:77_1_1_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  : (  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">Z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D11" title="MEMBERED:def.11">MEMBERED:def 11</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="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  : (  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">Z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span>}</span>  )</span><br/>

<a NAME="E1:77_1_1_1_2_1"/><i><font color="Green" title="E56">F</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>assume </b><a NAME="E2:77_1_1_1_2_1"/><i><font color="Green" title="E57">Z</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  : (  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">Z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span>}</span> </span><br/>

<a NAME="E3:77_1_1_1_2_1"/><b>then </b>
(  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">Z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c2">Z</font> )
 
<b>by </b><i><a class="ref" href="xxreal_2.html#T3" target="_self" title="XXREAL_2:th.3">Th3</a>, <a class="ref" href="xxreal_2.html#T4" target="_self" title="XXREAL_2:th.4">Th3B</a></i>;<br/>
<b>hence </b><a NAME="E4:77_1_1_1_2_1"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  : (  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">Z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E2:77_1_1_1_2_1"><i><font color="Green" title="E57">Z</font></i></a>, <a class="txt" href="#E1:77_1_1_1_2_1"><i><font color="Green" title="E56">F</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:77_1_1_1_2"/>
<font color="Maroon" title="c1">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="#E4:77_1_1_1_2"><i><font color="Green" title="E55">A</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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