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<b>let </b><font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">x</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> implies  ex <font color="Olive" title="b1">z</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>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Olive" title="b1">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Olive" title="b1">z</font> ) )</span><br/>



<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font>
 ; <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">z</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>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Olive" title="b1">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Olive" title="b1">z</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) or ( not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) or ( not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:43_1_1"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E2:43_1_1"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">  ex <font color="Olive" title="b1">z</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>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Olive" title="b1">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c3">y</font> as    <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>by </b><i><a class="txt" href="#E1:43_1_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="#E2:43_1_1"><i><font color="Green" title="E42">A3</font></i></a></i>;<br/><a NAME="E4:43_1_1"/>
<font color="Maroon" title="c6">y'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">x'</font>
 <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green" title="E18">Lm14</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">z'</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="E6:43_1_1"/><i><font color="Green" title="E43">A4</font></i>: 
( <font color="Maroon" title="c6">y'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">z'</font> &amp; <font color="Maroon" title="c7">z'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">x'</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T101" title="ARYTM_3:th.101">ARYTM_3:101</a></i>;<br/><a NAME="E7:43_1_1"/>
<font color="Maroon" title="c7">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">z'</font> as    <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>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self" title="ARYTM_2:th.1">Th1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">z</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c8">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font> )</span><br/><b>thus </b><a NAME="E9:43_1_1"/>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c8">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="txt" href="#E6:43_1_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green" title="E18">Lm14</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="43_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:43_1_2"/><i><font color="Green" title="E41">A5</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E2:43_1_2"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">  ex <font color="Olive" title="b1">z</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>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Olive" title="b1">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><a NAME="E3:43_1_2"/>
<font color="Maroon" title="c4">x</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/><a NAME="E4:43_1_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : for <font color="Olive" title="b2">r</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 <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>( ( for <font color="Olive" title="b3">s</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 <font color="Olive" title="b3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) &amp;  ex <font color="Olive" title="b3">s</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="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b3">s</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:43_1_2"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="arytm_2.html#E7"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E6:43_1_2"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 <b>and </b><br/><a NAME="E7:43_1_2"/><i><font color="Green" title="E44">A8</font></i>: 
for <font color="Olive" title="b1">r</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 <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/>( ( for <font color="Olive" title="b2">s</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 <font color="Olive" title="b2">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> ) &amp;  ex <font color="Olive" title="b2">s</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="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b2">s</font> ) )
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c3">y</font> as    <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>by </b><i><a class="txt" href="#E2:43_1_2"><i><font color="Green" title="E42">A6</font></i></a></i>;<br/><a NAME="E9:43_1_2"/>
<font color="Maroon" title="c6">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="#E1:43_1_2"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">s</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="E11:43_1_2"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 <b>and </b><br/><a NAME="E12:43_1_2"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c6">y'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="#E6:43_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E7:43_1_2"><i><font color="Green" title="E44">A8</font></i></a></i>;<br/><a NAME="E13:43_1_2"/>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">s</font> as    <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>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self" title="ARYTM_2:th.1">Th1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">z</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c8">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font> )</span><br/><b>thus </b><a NAME="E15:43_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c8">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font> )</span><br/><b>thus </b><a NAME="E16:43_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:43_1_2"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E6:43_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E11:43_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</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="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font></span><br/><b>thus </b><a NAME="E17:43_1_2"/>
<font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="#E12:43_1_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green" title="E18">Lm14</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="43_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:43_1_3"/><i><font color="Green" title="E41">A11</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E2:43_1_3"/><i><font color="Green" title="E42">A12</font></i>: 
not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">  ex <font color="Olive" title="b1">z</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>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Olive" title="b1">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><a NAME="E3:43_1_3"/>
<font color="Maroon" title="c3">y</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/><a NAME="E4:43_1_3"/><b>then </b>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : for <font color="Olive" title="b2">r</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 <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>( ( for <font color="Olive" title="b3">s</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 <font color="Olive" title="b3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) &amp;  ex <font color="Olive" title="b3">s</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="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b3">s</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E2:43_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="arytm_2.html#E7"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E6:43_1_3"/><i><font color="Green" title="E43">A13</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E7:43_1_3"/><i><font color="Green" title="E44">A14</font></i>: 
for <font color="Olive" title="b1">r</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 <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> holds <br/>( ( for <font color="Olive" title="b2">s</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 <font color="Olive" title="b2">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> ) &amp;  ex <font color="Olive" title="b2">s</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="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b2">s</font> ) )
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c4">x</font> as    <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>by </b><i><a class="txt" href="#E1:43_1_3"><i><font color="Green" title="E41">A11</font></i></a></i>;<br/><a NAME="E9:43_1_3"/><i><font color="Green" title="E45">A15</font></i>: 
not <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="#E2:43_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</a></i>;<br/><a NAME="E10:43_1_3"/>
<font color="Maroon" title="c5">B</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> 
 <b>by </b><i><a class="txt" href="#E2:43_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="#E6:43_1_3"><i><font color="Green" title="E43">A13</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">y1</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="E12:43_1_3"/><i><font color="Green" title="E46">A16</font></i>: 
<font color="Maroon" title="c7">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E13:43_1_3"/>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c7">y1</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></i>;<br/><a NAME="E14:43_1_3"/><b>then </b><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c6">x'</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> 
 <b>by </b><i><a class="txt" href="#E6:43_1_3"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E7:43_1_3"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="#E9:43_1_3"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="#E12:43_1_3"><i><font color="Green" title="E46">A16</font></i></a></i>;<br/><a NAME="E15:43_1_3"/>
 ex <font color="Olive" title="b1">z'</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="Olive" title="b1">z'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> &amp; not <font color="Olive" title="b1">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_3_1"/><i><font color="Green" title="E48">A18</font></i>: 
for <font color="Olive" title="b1">z'</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>   holds <br/>(  not <font color="Olive" title="b1">z'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> or <font color="Olive" title="b1">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>set </b><font color="Maroon" title="c8">C</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> </span>}</span> ;<br/>
<a NAME="E2:43_1_3_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1_3_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:43_1_3_1_1_1"/><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> </span>}</span> </span><br/>

<b>then reconsider </b><font color="Maroon" title="c10">z'</font> = <font color="Maroon" title="c9">e</font> as    <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>by </b><i><a class="txt" href="#E6:43_1_3"><i><font color="Green" title="E43">A13</font></i></a></i>;<br/>
<a NAME="E3:43_1_3_1_1_1"/>
not <font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">z'</font>
 
<b>by </b><i><a class="txt" href="#E6:43_1_3"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E7:43_1_3"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="#E9:43_1_3"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="#E1:43_1_3_1_1_1"><i><font color="Green" title="E49">A19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:43_1_3_1_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> </span>}</span> 
 ; <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>

<b>let </b><font color="Maroon" title="c9">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> </span>}</span>  or <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> )</span><br/>

<b>assume </b><a NAME="E2:43_1_3_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">s</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="E4:43_1_3_1_1"/><i><font color="Green" title="E49">A20</font></i>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font>
 <b>and </b><br/><a NAME="E5:43_1_3_1_1"/><i><font color="Green" title="E50">A21</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font>
 ;<br/>
<b>thus </b><a NAME="E6:43_1_3_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E1:43_1_3_1"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="#E4:43_1_3_1_1"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="#E5:43_1_3_1_1"><i><font color="Green" title="E50">A21</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>
<a NAME="E3:43_1_3_1"/><b>then </b>
<font color="Maroon" title="c3">y</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="b2">s</font> where <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#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</font> is    <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>  : <font color="Olive" title="b1">t</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>}</span> 
 
<b>by </b><i><a class="txt" href="#E14:43_1_3"><i><font color="Green" title="E47">A17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:43_1_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_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><b>then consider </b><font color="Maroon" title="c8">z'</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="E17:43_1_3"/><i><font color="Green" title="E48">A22</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font>
 <b>and </b><br/><a NAME="E18:43_1_3"/><i><font color="Green" title="E49">A23</font></i>: 
not <font color="Maroon" title="c8">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font>
 ;<br/><a NAME="E19:43_1_3"/>
<font color="Maroon" title="c8">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">z'</font> as    <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>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self" title="ARYTM_2:th.1">Th1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c9">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c9">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c9">z</font> )</span><br/><b>thus </b><a NAME="E21:43_1_3"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c9">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c9">z</font> )</span><br/><b>thus </b><a NAME="E22:43_1_3"/>
<font color="Maroon" title="c9">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E17:43_1_3"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green" title="E18">Lm14</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="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c9">z</font></span><br/><b>thus </b><a NAME="E23:43_1_3"/>
<font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c9">z</font>
 <b>by </b><i><a class="txt" href="#E2:43_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="#E18:43_1_3"><i><font color="Green" title="E49">A23</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</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="43_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:43_1_4"/><i><font color="Green" title="E41">A24</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E2:43_1_4"/><i><font color="Green" title="E42">A25</font></i>: 
not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">  ex <font color="Olive" title="b1">z</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>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Olive" title="b1">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><a NAME="E3:43_1_4"/><i><font color="Green" title="E43">A26</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="#E1:43_1_4"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="#E2:43_1_4"><i><font color="Green" title="E42">A25</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</a></i>;<br/><a NAME="E4:43_1_4"/>
<font color="Maroon" title="c4">x</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/><a NAME="E5:43_1_4"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : for <font color="Olive" title="b2">r</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 <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>( ( for <font color="Olive" title="b3">s</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 <font color="Olive" title="b3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) &amp;  ex <font color="Olive" title="b3">s</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="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b3">s</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:43_1_4"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="arytm_2.html#E7"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E7:43_1_4"/><i><font color="Green" title="E44">A27</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 <b>and </b><br/><a NAME="E8:43_1_4"/>
for <font color="Olive" title="b1">r</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 <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/>( ( for <font color="Olive" title="b2">s</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 <font color="Olive" title="b2">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> ) &amp;  ex <font color="Olive" title="b2">s</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="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b2">s</font> ) )
 ;<br/><a NAME="E9:43_1_4"/>
<font color="Maroon" title="c3">y</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/><a NAME="E10:43_1_4"/><b>then </b>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : for <font color="Olive" title="b2">r</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 <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>( ( for <font color="Olive" title="b3">s</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 <font color="Olive" title="b3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) &amp;  ex <font color="Olive" title="b3">s</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="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b3">s</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E2:43_1_4"><i><font color="Green" title="E42">A25</font></i></a>, <a class="txt" href="arytm_2.html#E7"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E12:43_1_4"/><i><font color="Green" title="E45">A28</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 <b>and </b><br/><a NAME="E13:43_1_4"/>
for <font color="Olive" title="b1">r</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 <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> holds <br/>( ( for <font color="Olive" title="b2">s</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 <font color="Olive" title="b2">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> ) &amp;  ex <font color="Olive" title="b2">s</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="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b2">s</font> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c7">z'</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="E15:43_1_4"/><i><font color="Green" title="E46">A29</font></i>: 
<font color="Maroon" title="c7">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 <b>and </b><br/><a NAME="E16:43_1_4"/><i><font color="Green" title="E47">A30</font></i>: 
not <font color="Maroon" title="c7">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="#E3:43_1_4"><i><font color="Green" title="E43">A26</font></i></a>, <a class="txt" href="#E7:43_1_4"><i><font color="Green" title="E44">A27</font></i></a>, <a class="txt" href="#E12:43_1_4"><i><font color="Green" title="E45">A28</font></i></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/><a NAME="E17:43_1_4"/>
<font color="Maroon" title="c7">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">z'</font> as    <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>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self" title="ARYTM_2:th.1">Th1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">z</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c8">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font> )</span><br/><b>thus </b><a NAME="E19:43_1_4"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c8">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font> )</span><br/><b>thus </b><a NAME="E20:43_1_4"/>
<font color="Maroon" title="c8">z</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:43_1_4"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="#E7:43_1_4"><i><font color="Green" title="E44">A27</font></i></a>, <a class="txt" href="#E15:43_1_4"><i><font color="Green" title="E46">A29</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</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="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font></span><br/><b>thus </b><a NAME="E21:43_1_4"/>
<font color="Maroon" title="c3">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="#E2:43_1_4"><i><font color="Green" title="E42">A25</font></i></a>, <a class="txt" href="#E12:43_1_4"><i><font color="Green" title="E45">A28</font></i></a>, <a class="txt" href="#E16:43_1_4"><i><font color="Green" title="E47">A30</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</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>

</div>
