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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2"/><i><font color="Green">E21</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>and </b><br/><a NAME="E2:2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<a NAME="E4:2"/>
<font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E7:2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E8:2"/><i><font color="Green">E25</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D4">XCMPLX_0:def 4</a></i>;<br/>
<a NAME="E9:2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
;<br/>
<a NAME="E10:2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/>
<a NAME="E11:2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>
<a NAME="E12:2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/>
<a NAME="E13:2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>
<a NAME="E14:2"/>
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/>
<a NAME="E15:2"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E3:2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E4:2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E5:2_1"/><i><font color="Green">E41</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a>, <a class="ref" href="arytm_2.html#T2">ARYTM_2:2</a></i>;<br/><a NAME="E6:2_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="arytm_2.html#T6">ARYTM_2:6</a></i>;<br/><b>hence </b><a NAME="E7:2_1"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K2">REAL+</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_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</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="E19:2"/><b>then </b>
<font color="Maroon">x</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_2.html#K2">REAL+</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>then consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E21:2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E22:2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E23:2"/><i><font color="Green">E45</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , , , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a>, <a class="ref" href="arytm_2.html#T2">ARYTM_2:2</a></i>;<br/>
<a NAME="E24:2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#T6">ARYTM_0:6</a></i>;<br/>
<a NAME="E25:2"/>
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="E26:2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</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="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E27:2"/>
not <font color="Maroon">r</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/>
<b>hence </b><a NAME="E28:2"/>
<font color="Maroon">x</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="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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