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<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:4"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:4"/><b>then </b><i><font color="Green">E7</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:4"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <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/>
<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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E1:4"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
0

;<br/><a NAME="E3:4_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><a NAME="E4:4_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><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="arytm_3.html#K7">RAT+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:4_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E3:4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="numbers.html#D3">NUMBERS:def 3</a>, <a class="ref" href="xboole_0.html#D4">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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:4_1_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E4:4_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rat_1.html#E3:4_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="rat_1.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E5:4_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><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="arytm_3.html#K7">RAT+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:4_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E3:4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="numbers.html#D3">NUMBERS:def 3</a>, <a class="ref" href="xboole_0.html#D4">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">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:4_1_3"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:4_1_3"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E4:4"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rat_1.html#E2:4_1_3"><i><font color="Green">E10</font></i></a>, <a class="ref" href="int_1.html#D1">INT_1:def 1</a></i>;<br/><a NAME="E5:4_1_3"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rat_1.html#E4:4_1_3"><i><font color="Green">E11</font></i></a></i>;<br/><a NAME="E6:4_1_3"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E7:4_1_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T1">ARYTM_2:1</a></i>;<br/><a NAME="E9:4_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E2:4_1_3"><i><font color="Green">E10</font></i></a></i>;<br/><a NAME="E10:4_1_3"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E1:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rat_1.html#E1:4_1_3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="rat_1.html#E5:4_1_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xcmplx_1.html#T50">XCMPLX_1:50</a></i>;<br/><i><font color="Green">E15</font></i>: <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="rat_1.html#E5:4_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rat_1.html#E5"><i><font color="Green">Lemma3</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="rat_1.html#E10:4_1_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="arytm_1.html#T19">ARYTM_1:19</a></i>
;<br/><a NAME="E12:4_1_3"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:4_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><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="arytm_3.html#K7">RAT+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="rat_1.html#E6:4_1_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rat_1.html#E11:4_1_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E14:4_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><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="arytm_3.html#K7">RAT+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E15:4_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E3:4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="numbers.html#D3">NUMBERS:def 3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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