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<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> ;<br/>



<b>assume </b><a NAME="E1:2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>suppose </b></a><a NAME="E1:2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="arytm_1.html#T18">ARYTM_1:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2"><b>suppose </b></a><a NAME="E1:2_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E2:2_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="arytm_1.html#T6">ARYTM_1:6</a></i>;<br/><a NAME="E3:2_1_2"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:2_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="arytm_1.html#T4">ARYTM_1:4</a></i>;<br/><a NAME="E4:2_1_2"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="arytm_3.html#K13">{}</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E2:2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E4:2_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rat_1.html#E1:2_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rat_1.html#E2:2_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="arytm_1.html#T10">ARYTM_1:10</a></i>;<br/></div><b>end;</b></div><a NAME="E6:2_1_2"/><b>then </b><i><font color="Green">E8</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:2_1_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="E8:2_1_2"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</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_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="rat_1.html#E4:2_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:2_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</sub></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="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="rat_1.html#E6:2_1_2"><i><font color="Green">E8</font></i></a>, <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>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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="E12:2_1_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E13:2_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E14:2_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> <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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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="E15:2_1_2"/>
<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="E16:2_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E12:2_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E17:2_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E12:2_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E18:2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E13:2_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E19:2_1_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E13:2_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E20:2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</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="E21:2_1_2"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E14:2_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E22:2_1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E14:2_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></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="E24:2_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E25:2_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E26:2_1_2"/><i><font color="Green">E18</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="rat_1.html#E8:2_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="rat_1.html#E17:2_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rat_1.html#E19:2_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><a NAME="E27:2_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E4:2_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rat_1.html#E19:2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="rat_1.html#E25:2_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><font color="Maroon">c<sub>3</sub></font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="arytm_2.html#D8">ARYTM_2:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="rat_1.html#E2:2_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="arytm_1.html#D1">ARYTM_1:def 1</a></i>
;<br/><a NAME="E29:2_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E1:2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="rat_1.html#E17:2_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rat_1.html#E22:2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="rat_1.html#E24:2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rat_1.html#E26:2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rat_1.html#E27:2_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="arytm_1.html#T18">ARYTM_1:18</a></i>;<br/><b>hence </b><a NAME="E30:2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
