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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 );<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E1:37_1_1"/><i><font color="Green">E19</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span>;<br/>
<a NAME="E2:37_1_1"/>
not  <a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span>
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T22">SCMFSA_2:22</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:37_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="scmfsa_2.html#K2">Int-Locations</a>  &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="scmfsa_2.html#M1">Int-Location</a>  <b>by </b><i><a class="txt" href="sfmastr1.html#E4:37_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="scmfsa_2.html#T11">SCMFSA_2:11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:37_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T19">SCMFSA_2:19</a></i>;<br/>
<a NAME="E8:37_1_1"/><i><font color="Green">E22</font></i>: 
( not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span> )
 
<b>by </b><i><a class="txt" href="sfmastr1.html#E4:37_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="sfmastr1.html#E7:37_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:37_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:37_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="sfmastr1.html#E8:37_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="sfmastr1.html#E1:37_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ,  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ] means for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>10</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>, <a href="subset_1.html#K1">{}</a> <a href="numbers.html#K5">NAT</a> ]
 ;<br/><b>hence </b><a NAME="E3:37_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2"/>
not <font color="Maroon">c<sub>10</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E3:37_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>hence </b><a NAME="E4:37_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:37_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:37_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E15:37_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="sfmastr1.html#E12:37_1_1"><i><font color="Green">E23</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E16:37_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="sfmastr1.html#E14:37_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>

<b>thus </b><a NAME="E17:37_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sfmastr1.html#E15:37_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means not <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>;<br/>
<a NAME="E18:37_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:37_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="sfmastr1.html#E14:37_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="scmfsa_2.html#M1">Int-Location</a>  =  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> ;<br/>
<a NAME="E3:37_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="sfmastr1.html#E2:37_1_1_2"><i><font color="Green">E27</font></i></a>);<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1_1_2_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 </span><b>by </b><i><a class="txt" href="sfmastr1.html#E1:37_1_1_2_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="scmfsa_2.html#K2">Int-Locations</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:37_1_1_2_1_1_1_2"/><i><font color="Green">E29</font></i>: 
(  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:37_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="scmfsa_2.html#K2">Int-Locations</a> 
 <b>by </b><i><a class="txt" href="sfmastr1.html#E3:37_1_1_2_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="scmfsa_2.html#T9">SCMFSA_2:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:37_1_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="scmfsa_2.html#K2">Int-Locations</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="scmfsa_2.html#M1">Int-Location</a>  <b>by </b><i><a class="ref" href="scmfsa_2.html#T11">SCMFSA_2:11</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:37_1_1_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T19">SCMFSA_2:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_2_1_2"/>
not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span>
 ;<br/><a NAME="E2:37_1_1_2_1_2"/><b>then </b>
( not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:37_1_1_2_1_2"/><b>then </b>
( not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="sfmastr1.html#E5:37_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:37_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>hence </b><a NAME="E5:37_1_1_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="sfmastr1.html#E5:37_1_1_2_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:37_1_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="sfmastr1.html#K1" target="_self">{</a><span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span><a href="sfmastr1.html#K1" target="_self">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:37_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T22">SCMFSA_2:22</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:37_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:37_1_1_3"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="sfmastr1.html#E21"><i><font color="Green">Lemma18</font></i></a></i>;<br/>
<a NAME="E3:37_1_1_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="sfmastr1.html#E15:37_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E4:37_1_1_3"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E5:37_1_1_3"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>assume </b><a NAME="E6:37_1_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E8:37_1_1_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<b>hence </b><a NAME="E9:37_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr1.html#E1:37_1_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sfmastr1.html#E3:37_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="sfmastr1.html#E5:37_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E20:37_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="sfmastr1.html#E18:37_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="sfmastr1.html#E19:37_1_1"><i><font color="Green">E27</font></i></a>);</i><br/>


</div>
