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<div class="add">

<a NAME="E1:4"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <a href="mesfunc1.html#K1" target="_self">INT-</a>  <a href="subset_1.html#K4">\/</a> <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="int_1.html#D1">INT_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="mesfunc1.html#K1" target="_self">INT-</a>  <a href="subset_1.html#K4">\/</a> <a href="numbers.html#K5">NAT</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="4_1_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:4_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K1" target="_self">INT-</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:4_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="mesfunc1.html#K1" target="_self">INT-</a>  <a href="subset_1.html#K4">\/</a> <a href="numbers.html#K5">NAT</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><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="mesfunc1.html#K1" target="_self">INT-</a>  <a href="subset_1.html#K4">\/</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:4"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="numbers.html#K4">INT</a>  <a href="tarski.html#R1">c=</a> <a href="mesfunc1.html#K1" target="_self">INT-</a>  <a href="subset_1.html#K4">\/</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:4"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <a href="mesfunc1.html#K1" target="_self">INT-</a>  <a href="subset_1.html#K4">\/</a> <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="mesfunc1.html#K1" target="_self">INT-</a>  <a href="subset_1.html#K4">\/</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K1" target="_self">INT-</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 </span><b>by </b><i>, <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="4_2_1_1_1"><b>suppose </b></a><a NAME="E1:4_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K1" target="_self">INT-</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:4_2_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:4_2_1_1_1"/>
<font color="Maroon">x</font> <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#D1">INT_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1_2"><b>suppose </b></a><a NAME="E1:4_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_2_1_1_2"/>
<font color="Maroon">x</font> <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#D1">INT_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:4"/><b>then </b>
<a href="mesfunc1.html#K1" target="_self">INT-</a>  <a href="subset_1.html#K4">\/</a> <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:4"/>
 <a href="numbers.html#K4">INT</a>  <a href="hidden.html#R1">=</a> <a href="mesfunc1.html#K1" target="_self">INT-</a>  <a href="subset_1.html#K4">\/</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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