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<b>let </b><font color="Maroon" title="c3">r</font>, <font color="Maroon" title="c4">s</font>, <font color="Maroon" title="c5">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">t</font> implies  ex <font color="Olive" title="b1">t0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Olive" title="b1">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> ) )</span><br/>





<b>given </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><a NAME="E2:93"/><i><font color="Green" title="E79">A1</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">ARYTM_3:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Olive" title="b1">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:93_1"/>
( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  or <font color="Maroon" title="c4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1"><b>suppose </b></a><a NAME="E1:93_1_1"/>
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Olive" title="b1">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> )</span><br/><div class="add"><a NAME="E2:93_1_1"/><b>then </b><i><font color="Green" title="E80">A2</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T54" target="_self" title="ARYTM_3:th.54">Th54</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">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="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c5">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> )</span><br/><b>thus </b><a NAME="E3:93_1_1"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c5">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:93"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="arytm_3.html#E2:93_1_1"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="arytm_3.html#T70" target="_self" title="ARYTM_3:th.70">Th70</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="93_1_2"><b>suppose </b></a><a NAME="E1:93_1_2"/><i><font color="Green" title="E80">A3</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Olive" title="b1">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">t0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E3:93_1_2"/><i><font color="Green" title="E81">A4</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T61" target="_self" title="ARYTM_3:th.61">Th61</a></i>;<br/><b>consider </b><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#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E5:93_1_2"/><i><font color="Green" title="E82">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t1</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:93_1_2"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="arytm_3.html#T61" target="_self" title="ARYTM_3:th.61">Th61</a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">t0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c7">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> )</span><br/><b>thus </b><a NAME="E6:93_1_2"/>
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">t0</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E3:93_1_2"><i><font color="Green" title="E81">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font></span><br/><b>take </b>
<font color="Maroon" title="c8">t1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">ARYTM_3:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">t1</font></span><br/><a NAME="E7:93_1_2"/>
<font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">t1</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:93"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="arytm_3.html#E3:93_1_2"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="arytm_3.html#E5:93_1_2"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="arytm_3.html#T63" target="_self" title="ARYTM_3:th.63">Th63</a></i>;<br/><b>hence </b><a NAME="E8:93_1_2"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">t1</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:93_1_2"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="arytm_3.html#T62" target="_self" title="ARYTM_3:th.62">Th62</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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