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<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green" title="E17">A2</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c7">z1</font> = <font color="Maroon" title="c4">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="E4:15"/><i><font color="Green" title="E18">A3</font></i>: 
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">c</font> &amp;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="txt" href="xreal_1.html#E20"><i><font color="Green" title="E14">Lm14</font></i></a></i>;<br/>
<a NAME="E5:15"/>
not <font color="Maroon" title="c1">o</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E6:15"/><b>then </b><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:15"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:15"/><i><font color="Green" title="E20">A5</font></i>: 
not <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">c</font>
 ;<br/>

<a NAME="E8:15"/><b>then </b><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1"/>
( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ) or ( <font color="Maroon" title="c2">a</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ) or ( <font color="Maroon" title="c2">a</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c3">b</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 </span><b>by </b><i><a class="txt" href="xreal_1.html#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:15_1_1"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:15_1_1"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c8">a''</font>, <font color="Maroon" title="c9">b''</font> being    <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> such that </b><br/><a NAME="E4:15_1_1"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a''</font>
 <b>and </b><br/><a NAME="E5:15_1_1"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b''</font>
 <b>and </b><br/><a NAME="E6:15_1_1"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c8">a''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">b''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="xreal_1.html#E1:15_1_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="xreal_1.html#E2:15_1_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c11">c'</font> being    <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> such that </b><br/><a NAME="E8:15_1_1"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a'</font>
 <b>and </b><br/><a NAME="E9:15_1_1"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 <b>and </b><br/><a NAME="E10:15_1_1"/><i><font color="Green" title="E29">A14</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:15"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="xreal_1.html#E1:15_1_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">b'</font>, <font color="Maroon" title="c13">c''</font> being    <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> such that </b><br/><a NAME="E12:15_1_1"/><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b'</font>
 <b>and </b><br/><a NAME="E13:15_1_1"/><i><font color="Green" title="E31">A16</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c''</font>
 <b>and </b><br/><a NAME="E14:15_1_1"/><i><font color="Green" title="E32">A17</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c13">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:15"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="xreal_1.html#E2:15_1_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></i>;<br/><a NAME="E15:15_1_1"/>
<font color="Maroon" title="c10">a'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c11">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c12">b'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E4:15_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="xreal_1.html#E5:15_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="xreal_1.html#E6:15_1_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="xreal_1.html#E8:15_1_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="xreal_1.html#E12:15_1_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="arytm_1.html#T8" title="ARYTM_1:th.8">ARYTM_1:8</a></i>;<br/><b>hence </b><a NAME="E16:15_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E4:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="xreal_1.html#E7:15"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="xreal_1.html#E9:15_1_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="xreal_1.html#E10:15_1_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="xreal_1.html#E13:15_1_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="xreal_1.html#E14:15_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="xreal_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:15_1_2"/><i><font color="Green" title="E22">A18</font></i>: 
<font color="Maroon" title="c2">a</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E2:15_1_2"/><i><font color="Green" title="E23">A19</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c9">c'</font> being    <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> such that </b><br/><a NAME="E4:15_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c8">a'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E5:15_1_2"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c'</font>
 <b>and </b><br/><a NAME="E6:15_1_2"/><i><font color="Green" title="E24">A20</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c9">c'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c8">a'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:15"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="xreal_1.html#E8:15"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="xreal_1.html#E1:15_1_2"><i><font color="Green" title="E22">A18</font></i></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">b'</font>, <font color="Maroon" title="c11">c''</font> being    <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> such that </b><br/><a NAME="E8:15_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font>
 <b>and </b><br/><a NAME="E9:15_1_2"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c''</font>
 <b>and </b><br/><a NAME="E10:15_1_2"/><i><font color="Green" title="E25">A21</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c11">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:15"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="xreal_1.html#E2:15_1_2"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></i>;<br/><a NAME="E11:15_1_2"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">z1</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:15_1_2"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="xreal_1.html#E7"><i><font color="Green" title="E7">Lm7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:15_1_2"/><b>then </b>
( not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">z1</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="xreal_1.html#E10:15_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E13:15_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E4:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="xreal_1.html#E7:15"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:15_1_3"/><i><font color="Green" title="E22">A22</font></i>: 
<font color="Maroon" title="c2">a</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E2:15_1_3"/><i><font color="Green" title="E23">A23</font></i>: 
<font color="Maroon" title="c3">b</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c8">a''</font>, <font color="Maroon" title="c9">b''</font> being    <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> such that </b><br/><a NAME="E4:15_1_3"/><i><font color="Green" title="E24">A24</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c8">a''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E5:15_1_3"/><i><font color="Green" title="E25">A25</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c9">b''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:15_1_3"/><i><font color="Green" title="E26">A26</font></i>: 
<font color="Maroon" title="c9">b''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">a''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="xreal_1.html#E1:15_1_3"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="xreal_1.html#E2:15_1_3"><i><font color="Green" title="E23">A23</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c11">c'</font> being    <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> such that </b><br/><a NAME="E8:15_1_3"/><i><font color="Green" title="E27">A27</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c10">a'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E9:15_1_3"/><i><font color="Green" title="E28">A28</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 <b>and </b><br/><a NAME="E10:15_1_3"/><i><font color="Green" title="E29">A29</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c11">c'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c10">a'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:15"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="xreal_1.html#E8:15"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="xreal_1.html#E1:15_1_3"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">b'</font>, <font color="Maroon" title="c13">c''</font> being    <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> such that </b><br/><a NAME="E12:15_1_3"/><i><font color="Green" title="E30">A30</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c12">b'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E13:15_1_3"/><i><font color="Green" title="E31">A31</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c''</font>
 <b>and </b><br/><a NAME="E14:15_1_3"/><i><font color="Green" title="E32">A32</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c13">c''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c12">b'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:15"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="xreal_1.html#E8:15"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="xreal_1.html#E2:15_1_3"><i><font color="Green" title="E23">A23</font></i></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></i>;<br/><a NAME="E15:15_1_3"/><i><font color="Green" title="E33">A33</font></i>: 
<font color="Maroon" title="c10">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E4:15_1_3"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="xreal_1.html#E8:15_1_3"><i><font color="Green" title="E27">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E16:15_1_3"/><i><font color="Green" title="E34">A34</font></i>: 
<font color="Maroon" title="c12">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:15_1_3"><i><font color="Green" title="E25">A25</font></i></a>, <a class="txt" href="xreal_1.html#E12:15_1_3"><i><font color="Green" title="E30">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E17:15_1_3"/><i><font color="Green" title="E35">A35</font></i>: 
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">z1</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">z1</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="xreal_1.html#E10:15_1_3"><i><font color="Green" title="E29">A29</font></i></a>, <a class="txt" href="xreal_1.html#E14:15_1_3"><i><font color="Green" title="E32">A32</font></i></a>, <a class="txt" href="xreal_1.html#E7"><i><font color="Green" title="E7">Lm7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E18:15_1_3"/>
<font color="Maroon" title="c12">b'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c11">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">a'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:15_1_3"><i><font color="Green" title="E26">A26</font></i></a>, <a class="txt" href="xreal_1.html#E15:15_1_3"><i><font color="Green" title="E33">A33</font></i></a>, <a class="txt" href="xreal_1.html#E16:15_1_3"><i><font color="Green" title="E34">A34</font></i></a>, <a class="ref" href="arytm_1.html#T8" title="ARYTM_1:th.8">ARYTM_1:8</a></i>;<br/><b>hence </b><a NAME="E19:15_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E4:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="xreal_1.html#E7:15"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="xreal_1.html#E9:15_1_3"><i><font color="Green" title="E28">A28</font></i></a>, <a class="txt" href="xreal_1.html#E10:15_1_3"><i><font color="Green" title="E29">A29</font></i></a>, <a class="txt" href="xreal_1.html#E13:15_1_3"><i><font color="Green" title="E31">A31</font></i></a>, <a class="txt" href="xreal_1.html#E14:15_1_3"><i><font color="Green" title="E32">A32</font></i></a>, <a class="txt" href="xreal_1.html#E17:15_1_3"><i><font color="Green" title="E35">A35</font></i></a>, <a class="txt" href="xreal_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
