<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">F</font> be  <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <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"> for <font color="Olive" title="b1">f</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Olive" title="b1">f</font> <a href="member_1.html#K21" title="MEMBER_1:func.21">///</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b2">w</font></span>)</span> where <font color="Olive" title="b2">w</font> is    <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>  : <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span> </span><br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="member_1.html#K21" title="MEMBER_1:func.21">///</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <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>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span> </span><br/>



<b>thus </b><a NAME="E1:329"/>
<font color="Maroon" title="c2">f</font> <a href="member_1.html#K21" title="MEMBER_1:func.21">///</a> <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <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>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</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"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <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>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="member_1.html#K21" title="MEMBER_1:func.21">///</a> <font color="Maroon" title="c1">F</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="329_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="member_1.html#K21" title="MEMBER_1:func.21">///</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c3">e</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="Maroon" title="c2">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <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>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:329_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="member_1.html#K21" title="MEMBER_1:func.21">///</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">e</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="Maroon" title="c2">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <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>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c4">w1</font>, <font color="Maroon" title="c5">w2</font> being    <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> <b> such that </b><br/><a NAME="E3:329_1"/><i><font color="Green" title="E143">A1</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w1</font> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <font color="Maroon" title="c5">w2</font>
 <b>and </b><br/><a NAME="E4:329_1"/><i><font color="Green" title="E144">A2</font></i>: 
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E5:329_1"/><i><font color="Green" title="E145">A3</font></i>: 
<font color="Maroon" title="c5">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="member_1.html#K3" title="MEMBER_1:func.3">""</a> 
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">w3</font> being    <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> <b> such that </b><br/><a NAME="E7:329_1"/><i><font color="Green" title="E146">A4</font></i>: 
<font color="Maroon" title="c5">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w3</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> 
 <b>and </b><br/><a NAME="E8:329_1"/><i><font color="Green" title="E147">A5</font></i>: 
<font color="Maroon" title="c6">w3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="#E5:329_1"><i><font color="Green" title="E145">A3</font></i></a></i>;<br/>
<a NAME="E9:329_1"/><i><font color="Green" title="E148">A6</font></i>: 
<font color="Maroon" title="c4">w1</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">w3</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w1</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <font color="Maroon" title="c6">w3</font>
 
;<br/>
<a NAME="E10:329_1"/>
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="#E4:329_1"><i><font color="Green" title="E144">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:329_1"/>
<font color="Maroon" title="c3">e</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="Maroon" title="c2">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <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>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:329_1"><i><font color="Green" title="E143">A1</font></i></a>, <a class="txt" href="#E7:329_1"><i><font color="Green" title="E146">A4</font></i></a>, <a class="txt" href="#E8:329_1"><i><font color="Green" title="E147">A5</font></i></a>, <a class="txt" href="#E9:329_1"><i><font color="Green" title="E148">A6</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>

<b>let </b><font color="Maroon" title="c3">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="c3">e</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="Maroon" title="c2">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <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>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span>  or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="member_1.html#K21" title="MEMBER_1:func.21">///</a> <font color="Maroon" title="c1">F</font> )</span><br/>

<b>assume </b><a NAME="E2:329"/>
<font color="Maroon" title="c3">e</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="Maroon" title="c2">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <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>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="member_1.html#K21" title="MEMBER_1:func.21">///</a> <font color="Maroon" title="c1">F</font></span><br/>

<a NAME="E3:329"/><b>then </b>
 ex <font color="Olive" title="b1">w</font> being   <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>  st <br/>( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> )
 
;<br/>
<b>hence </b><a NAME="E4:329"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="member_1.html#K21" title="MEMBER_1:func.21">///</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="ref" href="member_1.html#T216" target="_self" title="MEMBER_1:th.216">Th216</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>
