<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a> ;<br/>

<b>thus </b><a NAME="E1:57"/>
<font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c5">C</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font></span>)</span>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c7">r0</font>, <font color="Maroon" title="c8">v0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E4:57_1"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c8">v0</font>
 <b>and </b><br/><a NAME="E5:57_1"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c7">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E6:57_1"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon" title="c8">v0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font>
 ;<br/>
<a NAME="E7:57_1"/>
<font color="Maroon" title="c8">v0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E6:57_1"><i><font color="Green" title="E58">A3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">s0</font>, <font color="Maroon" title="c10">t0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E9:57_1"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c8">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c10">t0</font>
 <b>and </b><br/><a NAME="E10:57_1"/><i><font color="Green" title="E60">A5</font></i>: 
<font color="Maroon" title="c9">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E11:57_1"/><i><font color="Green" title="E61">A6</font></i>: 
<font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 ;<br/>
<a NAME="E12:57_1"/>
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:57_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="arytm_2.html#E10:57_1"><i><font color="Green" title="E60">A5</font></i></a></i>;<br/>
<a NAME="E13:57_1"/><b>then </b><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c4">B</font>
 
;<br/>
<a NAME="E14:57_1"/>
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:57_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="arytm_2.html#E11:57_1"><i><font color="Green" title="E61">A6</font></i></a></i>;<br/>
<a NAME="E15:57_1"/><b>then </b><i><font color="Green" title="E63">A8</font></i>: 
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c5">C</font>
 
;<br/>
<a NAME="E16:57_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">s0</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4:57_1"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="arytm_2.html#E9:57_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></i>;<br/>
<a NAME="E17:57_1"/><b>then </b>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E13:57_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="arytm_2.html#E15:57_1"><i><font color="Green" title="E63">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:57_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c5">C</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:57"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c5">C</font></span>)</span>
 ;<br/>

<a NAME="E3:57"/><b>then </b>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c7">s1</font>, <font color="Maroon" title="c8">t1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E5:57"/><i><font color="Green" title="E56">A9</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s1</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c8">t1</font>
 <b>and </b><br/><a NAME="E6:57"/><i><font color="Green" title="E57">A10</font></i>: 
<font color="Maroon" title="c7">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E7:57"/><i><font color="Green" title="E58">A11</font></i>: 
<font color="Maroon" title="c8">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c5">C</font>
 ;<br/>
<a NAME="E8:57"/>
<font color="Maroon" title="c7">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E6:57"><i><font color="Green" title="E57">A10</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">r0</font>, <font color="Maroon" title="c10">s0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E10:57"/><i><font color="Green" title="E59">A12</font></i>: 
<font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">s0</font>
 <b>and </b><br/><a NAME="E11:57"/><i><font color="Green" title="E60">A13</font></i>: 
<font color="Maroon" title="c9">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E12:57"/><i><font color="Green" title="E61">A14</font></i>: 
<font color="Maroon" title="c10">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ;<br/>
<a NAME="E13:57"/>
<font color="Maroon" title="c8">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E7:57"><i><font color="Green" title="E58">A11</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">r0'</font>, <font color="Maroon" title="c12">t0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E15:57"/><i><font color="Green" title="E62">A15</font></i>: 
<font color="Maroon" title="c8">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r0'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">t0</font>
 <b>and </b><br/><a NAME="E16:57"/><i><font color="Green" title="E63">A16</font></i>: 
<font color="Maroon" title="c11">r0'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E17:57"/><i><font color="Green" title="E64">A17</font></i>: 
<font color="Maroon" title="c12">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2"/>
( <font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c11">r0'</font> or <font color="Maroon" title="c11">r0'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1"><b>suppose </b></a><a NAME="E1:57_2_1"/>
<font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c11">r0'</font>
 ;<br/><div class="add"><a NAME="E2:57_2_1"/><b>then </b>
<font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c11">r0'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">s0'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E4:57_2_1"/><i><font color="Green" title="E65">A18</font></i>: 
<font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">s0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r0'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">s0'</font>
 <b>and </b><br/><a NAME="E5:57_2_1"/><i><font color="Green" title="E66">A19</font></i>: 
<font color="Maroon" title="c13">s0'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></i>;<br/><a NAME="E6:57_2_1"/>
<font color="Maroon" title="c13">s0'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E12:57"><i><font color="Green" title="E61">A14</font></i></a>, <a class="txt" href="arytm_2.html#E5:57_2_1"><i><font color="Green" title="E66">A19</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E22">Lm17</font></i></a></i>;<br/><a NAME="E7:57_2_1"/><b>then </b>
<font color="Maroon" title="c13">s0'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E17:57"><i><font color="Green" title="E64">A17</font></i></a></i>;<br/><a NAME="E8:57_2_1"/><b>then </b><i><font color="Green" title="E67">A20</font></i>: 
<font color="Maroon" title="c13">s0'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font>
 ;<br/><a NAME="E9:57_2_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r0'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">s0'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">t0</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:57"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="arytm_2.html#E10:57"><i><font color="Green" title="E59">A12</font></i></a>, <a class="txt" href="arytm_2.html#E15:57"><i><font color="Green" title="E62">A15</font></i></a>, <a class="txt" href="arytm_2.html#E4:57_2_1"><i><font color="Green" title="E65">A18</font></i></a>, <a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></i>;<br/><a NAME="E10:57_2_1"/><b>then </b>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E16:57"><i><font color="Green" title="E63">A16</font></i></a>, <a class="txt" href="arytm_2.html#E8:57_2_1"><i><font color="Green" title="E67">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E11:57_2_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2"><b>suppose </b></a><a NAME="E1:57_2_2"/>
<font color="Maroon" title="c11">r0'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r0</font>
 ;<br/><div class="add"><a NAME="E2:57_2_2"/><b>then </b>
<font color="Maroon" title="c11">r0'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">t0'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E4:57_2_2"/><i><font color="Green" title="E65">A21</font></i>: 
<font color="Maroon" title="c11">r0'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">t0'</font>
 <b>and </b><br/><a NAME="E5:57_2_2"/><i><font color="Green" title="E66">A22</font></i>: 
<font color="Maroon" title="c13">t0'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c12">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></i>;<br/><a NAME="E6:57_2_2"/>
<font color="Maroon" title="c13">t0'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E17:57"><i><font color="Green" title="E64">A17</font></i></a>, <a class="txt" href="arytm_2.html#E5:57_2_2"><i><font color="Green" title="E66">A22</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E22">Lm17</font></i></a></i>;<br/><a NAME="E7:57_2_2"/><b>then </b>
<font color="Maroon" title="c10">s0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c13">t0'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E12:57"><i><font color="Green" title="E61">A14</font></i></a></i>;<br/><a NAME="E8:57_2_2"/><b>then </b><i><font color="Green" title="E67">A23</font></i>: 
<font color="Maroon" title="c10">s0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c13">t0'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font>
 ;<br/><a NAME="E9:57_2_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c13">t0'</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:57"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="arytm_2.html#E10:57"><i><font color="Green" title="E59">A12</font></i></a>, <a class="txt" href="arytm_2.html#E15:57"><i><font color="Green" title="E62">A15</font></i></a>, <a class="txt" href="arytm_2.html#E4:57_2_2"><i><font color="Green" title="E65">A21</font></i></a>, <a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></i>;<br/><a NAME="E10:57_2_2"/><b>then </b>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E11:57"><i><font color="Green" title="E60">A13</font></i></a>, <a class="txt" href="arytm_2.html#E8:57_2_2"><i><font color="Green" title="E67">A23</font></i></a></i>;<br/><b>hence </b><a NAME="E11:57_2_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
