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<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> implies  ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> ) )</span><br/>

<b>assume </b><a NAME="E1:20"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">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">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c2">x1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:20"/><i><font color="Green" title="E14">A1</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x1</font> &amp; <font color="Maroon" title="c2">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> )
 <b>by </b><i><a class="ref" href="measure6.html#T23" target="_self" title="MEASURE6:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">x2</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:20"/><i><font color="Green" title="E15">A2</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x2</font> &amp; <font color="Maroon" title="c2">x1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E3:20"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="measure6.html#T24" target="_self" title="MEASURE6:th.24">Th24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
( <font color="Maroon" title="c2">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x2</font> or <font color="Maroon" title="c3">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> )</span><br/><div class="add"><a NAME="E2:20_1_1"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T96" title="SUPINF_1:th.96">SUPINF_1:96</a></i>;<br/><b>thus </b><a NAME="E3:20_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="measure6.html#E3:20"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="measure6.html#E1:20_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="supinf_1.html#T96" title="SUPINF_1:th.96">SUPINF_1:96</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="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font></span><br/><a NAME="E4:20_1_1"/>
<font color="Maroon" title="c2">x1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="txt" href="measure6.html#E3:20"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="measure6.html#T15" target="_self" title="MEASURE6:th.15">Th15</a></i>;<br/><a NAME="E5:20_1_1"/><b>then </b>
( <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">x2</font> &amp; <font color="Maroon" title="c2">x1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E5:20"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="measure6.html#E1:20_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="measure6.html#E2:20_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E6:20_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="measure6.html#E5:20"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c3">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> )</span><br/><div class="add"><a NAME="E2:20_1_2"/><b>then </b><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T96" title="SUPINF_1:th.96">SUPINF_1:96</a></i>;<br/><b>thus </b><a NAME="E3:20_1_2"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="measure6.html#E5:20"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="measure6.html#E1:20_1_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="supinf_1.html#T96" title="SUPINF_1:th.96">SUPINF_1:96</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="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font></span><br/><a NAME="E4:20_1_2"/>
( <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">x2</font> &amp; <font color="Maroon" title="c2">x1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E5:20"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="measure6.html#E1:20_1_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="measure6.html#E2:20_1_2"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E5:20_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="measure6.html#E5:20"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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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