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<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">l</font>
 ;<br/>

<a NAME="E2:4"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:4"/><b>then </b><i><font color="Green" title="E7">A2</font></i>: 
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="numbers.html#D1" title="NUMBERS:def.1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:4"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c4">m</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> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_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"/>
( <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> or ( <font color="Maroon" title="c3">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>suppose </b></a><a NAME="E1:4_1_1"/>
<font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c4">m</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><a NAME="E3:4_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="rat_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:4_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="numbers.html#D3" title="NUMBERS:def.3">NUMBERS:def 3</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><a NAME="E1:4_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:4_1_2"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 ;<br/><a NAME="E4:4_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#NK9" title="ARYTM_3:NK.9">quotient</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c3">l</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="rat_1.html#E3:4_1_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="rat_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/><a NAME="E5:4_1_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:4_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="numbers.html#D3" title="NUMBERS:def.3">NUMBERS:def 3</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:4_1_3"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c3">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:4_1_3"/><i><font color="Green" title="E10">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:4_1_3"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E4:4"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="rat_1.html#E2:4_1_3"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="int_1.html#D1" title="INT_1:def.1">INT_1:def 1</a></i>;<br/><a NAME="E5:4_1_3"/><i><font color="Green" title="E12">A8</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="rat_1.html#E4:4_1_3"><i><font color="Green" title="E11">A7</font></i></a></i>;<br/><a NAME="E6:4_1_3"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#NK9" title="ARYTM_3:NK.9">quotient</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c3">l</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/><a NAME="E7:4_1_3"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c5">k</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T1" title="ARYTM_2:th.1">ARYTM_2:1</a></i>;<br/><a NAME="E9:4_1_3"/>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E2:4_1_3"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/><a NAME="E10:4_1_3"/><b>then </b><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c6">x'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="rat_1.html#E1:4_1_3"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="rat_1.html#E5:4_1_3"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="xcmplx_1.html#T50" title="XCMPLX_1:th.50">XCMPLX_1:50</a></i>;<br/><i><font color="Green" title="E15">A11</font></i>: <font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c1">0'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c6">x'</font>
<b>by </b><i><a class="txt" href="rat_1.html#E5:4_1_3"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="rat_1.html#E5"><i><font color="Green" title="E3">Lm3</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c6">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="rat_1.html#E10:4_1_3"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="arytm_1.html#T19" title="ARYTM_1:th.19">ARYTM_1:19</a></i>
;<br/><a NAME="E12:4_1_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E13:4_1_3"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="rat_1.html#E6:4_1_3"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="rat_1.html#E11:4_1_3"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E14:4_1_3"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E15:4_1_3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="numbers.html#D3" title="NUMBERS:def.3">NUMBERS:def 3</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
