<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span><br/>

<a NAME="E1:45"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">A</font> iff <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">A</font> iff <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a><span class="kw"> such that </span><br/><a NAME="E2:45_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="membered.html#T8" onmouseover="rs('membered/T8')" onmouseout="rh()">MEMBERED:8</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">A</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:45_1_1"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><a NAME="E2:45_1_1"/><span class="kw">then </span>
 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="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <br/>( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Olive" title="b1">z</font> &amp; <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="member_1.html#T188" onmouseover="rs('member_1/T188')" onmouseout="rh()">MEMBER_1:188</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_1_1"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:45_1"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E5:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">r</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:45_1"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="member_1.html#T186" onmouseover="rs('member_1/T186')" onmouseout="rh()">MEMBER_1:186</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:45"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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