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<b>thus </b><a NAME="E1:14"/>
 <a href="numbers.html#K4">INT</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K2" target="_self">SCM+FSA-Data*-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K3" target="_self">SCM+FSA-Instr-Loc</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:14_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is  not  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E3:14_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:14_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K4">INT</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="scmfsa_1.html#E1:14_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:14_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K2" target="_self">SCM+FSA-Data*-Loc</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:14_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K2" target="_self">SCM+FSA-Data*-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K3" target="_self">SCM+FSA-Instr-Loc</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:14_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:14_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K3" target="_self">SCM+FSA-Instr-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K2" target="_self">SCM+FSA-Data*-Loc</a> 
 <b>by </b><i><a class="ref" href="ami_3.html#T4">AMI_3:4</a>, <a class="ref" href="ami_3.html#D1">AMI_3:def 1</a>, <a class="ref" href="ami_5.html#T23">AMI_5:23</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K2" target="_self">SCM+FSA-Data*-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K3" target="_self">SCM+FSA-Instr-Loc</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<a NAME="E2:14"/>
0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E3:14"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E4:14"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E5:14"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K2" target="_self">SCM+FSA-Data*-Loc</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:14"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K2" target="_self">SCM+FSA-Data*-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="scmfsa_1.html#K3" target="_self">SCM+FSA-Instr-Loc</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


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