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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">r</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"> ( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/>

<a NAME="E3:8"/><i><font color="Green" title="E11">A5</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T26" title="SUPINF_2:th.26">SUPINF_2:26</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">W</font> = <font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font>;<br/>
<a NAME="E4:8"/><i><font color="Green" title="E12">A8</font></i>: 
(  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 
<b>by </b><i><a class="txt" href="#E2:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="extreal1.html#T4" title="EXTREAL1:th.4">EXTREAL1:4</a></i>;<br/>
<a NAME="E5:8"/><i><font color="Green" title="E13">A61</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E14">A60</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/><b>then reconsider </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><a NAME="E3:8_1"/><i><font color="Green" title="E15">A62</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="#E5:8"><i><font color="Green" title="E13">A61</font></i></a>, <a class="txt" href="#E1:8_1"><i><font color="Green" title="E14">A60</font></i></a>, <a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>;<br/><a NAME="E4:8_1"/><i><font color="Green" title="E16">A63</font></i>: 
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E5:8"><i><font color="Green" title="E13">A61</font></i></a>, <a class="ref" href="supinf_2.html#D6" title="SUPINF_2:def.6">SUPINF_2:def 6</a></i>;<br/><a NAME="E5:8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="#E2:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:8"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="extreal1.html#T8" title="EXTREAL1:th.8">EXTREAL1:8</a></i>;<br/><a NAME="E6:8_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T9" title="EXTREAL1:th.9">EXTREAL1:9</a></i>;<br/><a NAME="E7:8_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/><b>hence </b><a NAME="E8:8_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:8_1"><i><font color="Green" title="E15">A62</font></i></a>, <a class="txt" href="#E4:8_1"><i><font color="Green" title="E16">A63</font></i></a>, <a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2: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>
<a NAME="E7:8"/><b>then </b><i><font color="Green" title="E14">A64</font></i>: 
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> implies <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )</span><br/><b>assume </b><a NAME="E1:8_2"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</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="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:8_2"/><i><font color="Green" title="E15">A65</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>;<br/><a NAME="E4:8_2"/>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:8_2"><i><font color="Green" title="E15">A65</font></i></a>, <a class="ref" href="supinf_2.html#T23" title="SUPINF_2:th.23">SUPINF_2:23</a></i>;<br/><a NAME="E5:8_2"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T22" title="SUPINF_2:th.22">SUPINF_2:22</a></i>;<br/><a NAME="E6:8_2"/><b>then </b><i><font color="Green" title="E16">A66</font></i>: 
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">y1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E8:8_2"/><i><font color="Green" title="E17">A67</font></i>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c9">y1</font> )
 <b>by </b><i><a class="txt" href="#E3:8_2"><i><font color="Green" title="E15">A65</font></i></a>, <a class="ref" href="supinf_2.html#D5" title="SUPINF_2:def.5">SUPINF_2:def 5</a></i>;<br/><a NAME="E9:8_2"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c9">y1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:8_2"><i><font color="Green" title="E15">A65</font></i></a>, <a class="txt" href="#E6:8_2"><i><font color="Green" title="E16">A66</font></i></a>, <a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E8:8_2"><i><font color="Green" title="E17">A67</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:8_2"/><b>then </b>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c9">y1</font>
 <b>by </b><i><a class="txt" href="#E2:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:8"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="extreal1.html#T8" title="EXTREAL1:th.8">EXTREAL1:8</a></i>;<br/><a NAME="E11:8_2"/><b>then </b>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c9">y1</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T9" title="EXTREAL1:th.9">EXTREAL1:9</a></i>;<br/><b>hence </b><a NAME="E12:8_2"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="#E8:8_2"><i><font color="Green" title="E17">A67</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</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="E9:8"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E10:8"/><b>then </b><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:8"><i><font color="Green" title="E14">A64</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:8"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:8"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="supinf_2.html#T26" title="SUPINF_2:th.26">SUPINF_2:26</a></i>;<br/>
<a NAME="E12:8"/><b>then </b>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T33" title="SUPINF_2:th.33">SUPINF_2:33</a></i>;<br/>
<a NAME="E13:8"/><b>then </b>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="xxreal_2.html#T13" title="XXREAL_2:th.13">XXREAL_2:13</a></i>;<br/>
<a NAME="E14:8"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="extreal2.html#T32" title="EXTREAL2:th.32">EXTREAL2:32</a></i>;<br/>
<a NAME="E15:8"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="xxreal_2.html#T11" title="XXREAL_2:th.11">XXREAL_2:11</a></i>;<br/>
<b>hence </b><a NAME="E16:8"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:8"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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>
