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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c2">G</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:290"/><span class="lab"><font color="Green" title="E123">A1</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/>

<a NAME="E2:290"/><span class="lab"><font color="Green" title="E124">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</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#K22" title="MEMBER_1:func.22">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="290_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">j</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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="membered.html#D8" title="MEMBERED:def.8">MEMBERED:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">j</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#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c2">G</font></span>)</span> or <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">G</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:290_1"/><span class="lab"><font color="Green" title="E125">A3</font></span>: 
<font color="Maroon" title="c4">j</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#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c2">G</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/>

<a NAME="E2:290_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E4:290_1"/><span class="lab"><font color="Green" title="E126">A4</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c5">w</font>
 <span class="kw">and </span><br/><a NAME="E5:290_1"/><span class="lab"><font color="Green" title="E127">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="member_1.html#T188" target="_self" title="MEMBER_1:th.188">Th188</a></span>;<br/>
<a NAME="E6:290_1"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:290_1"><span class="lab"><font color="Green" title="E125">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E8:290_1"/><span class="lab"><font color="Green" title="E128">A6</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c6">w1</font>
 <span class="kw">and </span><br/><a NAME="E9:290_1"/><span class="lab"><font color="Green" title="E129">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="member_1.html#T188" target="_self" title="MEMBER_1:th.188">Th188</a></span>;<br/>
<a NAME="E10:290_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:290"><span class="lab"><font color="Green" title="E123">A1</font></span></a>, <a class="txt" href="#E4:290_1"><span class="lab"><font color="Green" title="E126">A4</font></span></a>, <a class="txt" href="#E8:290_1"><span class="lab"><font color="Green" title="E128">A6</font></span></a>, <a class="ref" href="xxreal_3.html#T68" title="XXREAL_3:th.68">XXREAL_3:68</a></span>;<br/>
<a NAME="E11:290_1"/><span class="kw">then </span>
<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="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:290_1"><span class="lab"><font color="Green" title="E127">A5</font></span></a>, <a class="txt" href="#E9:290_1"><span class="lab"><font color="Green" title="E129">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:290_1"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:290_1"><span class="lab"><font color="Green" title="E126">A4</font></span></a>, <a class="ref" href="member_1.html#T186" target="_self" title="MEMBER_1:th.186">Th186</a></span>; <a class="txt" 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>
<a NAME="E3:290"/>
<font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c2">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="member_1.html#T83" target="_self" title="MEMBER_1:th.83">Th83</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:290"/>
<font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="member_1.html#K22" title="MEMBER_1:func.22">**</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:290"><span class="lab"><font color="Green" title="E124">A2</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
