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<div class="add">

<b>let </b><font color="Maroon" title="c8">r</font>, <font color="Maroon" title="c9">s</font>, <font color="Maroon" title="c10">t</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">t</font> implies <font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c8">r</font>, <font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c9">s</font>, <font color="Maroon" title="c13">z1</font> = <font color="Maroon" title="c10">t</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" target="_self" title="XREAL_0:def.1">Def1</a></i>;<br/>
<a NAME="E4:15"/>
for <font color="Olive" title="b1">x'</font> being   <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> <br/> for <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> holds <br/> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Olive" title="b1">x'</font>,<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x'</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> holds <br/> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font></span><br/><b>let </b><font color="Maroon" title="c15">r</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r</font> implies  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font> )</span><br/>



<b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font></span><br/>

<b>consider </b><font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c17">x2</font>, <font color="Maroon" title="c18">y1</font>, <font color="Maroon" title="c19">y2</font> being    <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> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E4:15_1"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c18">y1</font>,<font color="Maroon" title="c19">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E5:15_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c15">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c18">y1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c17">x2</font>,<font color="Maroon" title="c19">y2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c19">y2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c17">x2</font>,<font color="Maroon" title="c18">y1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5" title="XCMPLX_0:def.5">XCMPLX_0:def 5</a></i>;<br/>
<a NAME="E6:15_1"/><i><font color="Green" title="E26">A7</font></i>: 
( <font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y1</font> )
 
<b>by </b><i><a class="txt" href="#E3:15_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E4:15_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="xreal_0.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<a NAME="E7:15_1"/><i><font color="Green" title="E27">A8</font></i>: 
( <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c19">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E3:15_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E4:15_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="xreal_0.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<a NAME="E8:15_1"/><b>then </b>
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c19">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c17">x2</font>,<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b><i><font color="Green" title="E28">A9</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c19">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c17">x2</font>,<font color="Maroon" title="c18">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></i>;<br/>
<b>thus </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c13">z1</font> = 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c18">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c17">x2</font></span>)</span>,<font color="Maroon" title="c19">y2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E6:15_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="#E7:15_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a>, <a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c18">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c17">x2</font>,<font color="Maroon" title="c19">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17" title="ARYTM_0:th.17">ARYTM_0:17</a></i>
<br/>.= 
<font color="Maroon" title="c15">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font>
<b>by </b><i><a class="txt" href="#E5:15_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="#E9:15_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:15"/><b>then </b><i><font color="Green" title="E22">A10</font></i>: 
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c12">y1</font>,<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font> &amp;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font> )
 
;<br/>
<a NAME="E6:15"/>
not <font color="Maroon" title="c7">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"><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_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="E7:15"/><b>then </b><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c10">t</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="#E2:15"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/>
<b>assume </b><a NAME="E8:15"/><i><font color="Green" title="E24">A12</font></i>: 
not <font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E9:15"/><b>then </b><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2"/>
( ( <font color="Maroon" title="c8">r</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="c9">s</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="c8">r</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_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="c9">s</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="c8">r</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_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="c9">s</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_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="#E1:15"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1"><b>suppose </b></a><b>that </b><a NAME="E1:15_2_1"/><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c8">r</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_2_1"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c9">s</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c14">x''</font>, <font color="Maroon" title="c15">s''</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_2_1"/><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x''</font>
 <b>and </b><br/><a NAME="E5:15_2_1"/><i><font color="Green" title="E29">A17</font></i>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s''</font>
 <b>and </b><br/><a NAME="E6:15_2_1"/><i><font color="Green" title="E30">A18</font></i>: 
<font color="Maroon" title="c14">x''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c15">s''</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="txt" href="#E2:15_2_1"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">x'</font>, <font color="Maroon" title="c17">t'</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_2_1"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x'</font>
 <b>and </b><br/><a NAME="E9:15_2_1"/><i><font color="Green" title="E32">A20</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t'</font>
 <b>and </b><br/><a NAME="E10:15_2_1"/><i><font color="Green" title="E33">A21</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c17">t'</font>
 <b>by </b><i><a class="txt" href="#E7:15"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E26">A14</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="c18">s'</font>, <font color="Maroon" title="c19">t''</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_2_1"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s'</font>
 <b>and </b><br/><a NAME="E13:15_2_1"/><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">t''</font>
 <b>and </b><br/><a NAME="E14:15_2_1"/><i><font color="Green" title="E36">A24</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c12">y1</font>,<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c19">t''</font>
 <b>by </b><i><a class="txt" href="#E7:15"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="#E2:15_2_1"><i><font color="Green" title="E27">A15</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_2_1"/>
<font color="Maroon" title="c16">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c17">t'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c18">s'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c17">t'</font>
 <b>by </b><i><a class="txt" href="#E4:15_2_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="#E5:15_2_1"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="#E6:15_2_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="#E8:15_2_1"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="#E12:15_2_1"><i><font color="Green" title="E34">A22</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_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:15"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="#E9:15_2_1"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="#E10:15_2_1"><i><font color="Green" title="E33">A21</font></i></a>, <a class="txt" href="#E13:15_2_1"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="#E14:15_2_1"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="xreal_0.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2"><b>suppose </b></a><b>that </b><a NAME="E1:15_2_2"/><i><font color="Green" title="E26">A25</font></i>: 
<font color="Maroon" title="c8">r</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_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_2_2"/><i><font color="Green" title="E27">A26</font></i>: 
<font color="Maroon" title="c9">s</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c14">x'</font>, <font color="Maroon" title="c15">t'</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_2_2"/>
<font color="Maroon" title="c8">r</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E5:15_2_2"/>
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t'</font>
 <b>and </b><br/><a NAME="E6:15_2_2"/><i><font color="Green" title="E28">A27</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">t'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c14">x'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E7:15"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="#E9:15"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="#E1:15_2_2"><i><font color="Green" title="E26">A25</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="c16">s'</font>, <font color="Maroon" title="c17">t''</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_2_2"/>
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s'</font>
 <b>and </b><br/><a NAME="E9:15_2_2"/>
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t''</font>
 <b>and </b><br/><a NAME="E10:15_2_2"/><i><font color="Green" title="E29">A28</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c12">y1</font>,<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c17">t''</font>
 <b>by </b><i><a class="txt" href="#E7:15"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="#E2:15_2_2"><i><font color="Green" title="E27">A26</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_2_2"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">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"><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_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="#E6:15_2_2"><i><font color="Green" title="E28">A27</font></i></a>, <a class="txt" href="xreal_0.html#E4"><i><font color="Green" title="E4">Lm3</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_2_2"/><b>then </b>
( not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">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="c12">y1</font>,<font color="Maroon" title="c13">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"><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_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="#E10:15_2_2"><i><font color="Green" title="E29">A28</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_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:15"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_3"><b>suppose </b></a><b>that </b><a NAME="E1:15_2_3"/><i><font color="Green" title="E26">A29</font></i>: 
<font color="Maroon" title="c8">r</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_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_2_3"/><i><font color="Green" title="E27">A30</font></i>: 
<font color="Maroon" title="c9">s</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_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c14">x''</font>, <font color="Maroon" title="c15">s''</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_2_3"/><i><font color="Green" title="E28">A31</font></i>: 
<font color="Maroon" title="c8">r</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E5:15_2_3"/><i><font color="Green" title="E29">A32</font></i>: 
<font color="Maroon" title="c9">s</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c15">s''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:15_2_3"/><i><font color="Green" title="E30">A33</font></i>: 
<font color="Maroon" title="c15">s''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c14">x''</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E1:15_2_3"><i><font color="Green" title="E26">A29</font></i></a>, <a class="txt" href="#E2:15_2_3"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">x'</font>, <font color="Maroon" title="c17">t'</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_2_3"/><i><font color="Green" title="E31">A34</font></i>: 
<font color="Maroon" title="c8">r</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c16">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E9:15_2_3"/><i><font color="Green" title="E32">A35</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t'</font>
 <b>and </b><br/><a NAME="E10:15_2_3"/><i><font color="Green" title="E33">A36</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c17">t'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c16">x'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E7:15"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="#E9:15"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="#E1:15_2_3"><i><font color="Green" title="E26">A29</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="c18">s'</font>, <font color="Maroon" title="c19">t''</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_2_3"/><i><font color="Green" title="E34">A37</font></i>: 
<font color="Maroon" title="c9">s</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c18">s'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E13:15_2_3"/><i><font color="Green" title="E35">A38</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">t''</font>
 <b>and </b><br/><a NAME="E14:15_2_3"/><i><font color="Green" title="E36">A39</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c12">y1</font>,<font color="Maroon" title="c13">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c19">t''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c18">s'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E7:15"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="#E9:15"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="#E2:15_2_3"><i><font color="Green" title="E27">A30</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_2_3"/><i><font color="Green" title="E37">A40</font></i>: 
<font color="Maroon" title="c16">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x''</font>
 <b>by </b><i><a class="txt" href="#E4:15_2_3"><i><font color="Green" title="E28">A31</font></i></a>, <a class="txt" href="#E8:15_2_3"><i><font color="Green" title="E31">A34</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_2_3"/><i><font color="Green" title="E38">A41</font></i>: 
<font color="Maroon" title="c18">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s''</font>
 <b>by </b><i><a class="txt" href="#E5:15_2_3"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="#E12:15_2_3"><i><font color="Green" title="E34">A37</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_2_3"/><i><font color="Green" title="E39">A42</font></i>: 
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">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"><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_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="c12">y1</font>,<font color="Maroon" title="c13">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"><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_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="#E10:15_2_3"><i><font color="Green" title="E33">A36</font></i></a>, <a class="txt" href="#E14:15_2_3"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="xreal_0.html#E4"><i><font color="Green" title="E4">Lm3</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_2_3"/>
<font color="Maroon" title="c18">s'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c17">t'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c16">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c17">t'</font>
 <b>by </b><i><a class="txt" href="#E6:15_2_3"><i><font color="Green" title="E30">A33</font></i></a>, <a class="txt" href="#E15:15_2_3"><i><font color="Green" title="E37">A40</font></i></a>, <a class="txt" href="#E16:15_2_3"><i><font color="Green" title="E38">A41</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_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:15"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="#E9:15_2_3"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="#E10:15_2_3"><i><font color="Green" title="E33">A36</font></i></a>, <a class="txt" href="#E13:15_2_3"><i><font color="Green" title="E35">A38</font></i></a>, <a class="txt" href="#E14:15_2_3"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="#E17:15_2_3"><i><font color="Green" title="E39">A42</font></i></a>, <a class="txt" href="xreal_0.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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