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

<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</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">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Olive" title="b1">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">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="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">G</font></span>)</span> <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="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/>





<a NAME="E1:216"/><i><font color="Green" title="E99">A1</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c2">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="216_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D8" title="MEMBERED:def.8">MEMBERED:def 8</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="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">G</font></span>)</span> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c2">G</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:216_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">G</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="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c5">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> <b> such that </b><br/><a NAME="E3:216_1"/><i><font color="Green" title="E100">A2</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c5">w</font>
 <b>and </b><br/><a NAME="E4:216_1"/><i><font color="Green" title="E101">A3</font></i>: 
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="member_1.html#T140" target="_self" title="MEMBER_1:th.140">Th140</a></i>;<br/>
<a NAME="E5:216_1"/><i><font color="Green" title="E102">A4</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="#E4:216_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="member_1.html#T138" target="_self" title="MEMBER_1:th.138">Th138</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="216_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:216_1_1"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c6">w1</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="E3:216_1_1"/><i><font color="Green" title="E103">A5</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c6">w1</font>
 <b>and </b><br/><a NAME="E4:216_1_1"/><i><font color="Green" title="E104">A6</font></i>: 
<font color="Maroon" title="c6">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="member_1.html#T140" target="_self" title="MEMBER_1:th.140">Th140</a></i>;<br/><a NAME="E5:216_1_1"/>
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w1</font>
 <b>by </b><i><a class="txt" href="#E3:216_1_1"><i><font color="Green" title="E103">A5</font></i></a>, <a class="ref" href="member_1.html#T4" target="_self" title="MEMBER_1:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E6:216_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:216_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="#E4:216_1_1"><i><font color="Green" title="E104">A6</font></i></a>, <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>hence </b><a NAME="E7:216_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:216_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="#E5:216_1"><i><font color="Green" title="E102">A4</font></i></a>, <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>
<a NAME="E2:216"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="member_1.html#T144" target="_self" title="MEMBER_1:th.144">Th144</a></i>;<br/>
<b>hence </b><a NAME="E3:216"/>
<font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">G</font></span>)</span> <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="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K13" title="MEMBER_1:func.13">++</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:216"><i><font color="Green" title="E99">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
