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<a NAME="E1:15_1_1"/>
 <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_1_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="jordan1h.html#E1:15_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E3:15_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan1h.html#K1" target="_self">RealOrd</a> 
 
;<br/>
<b>hence </b><a NAME="E4:15_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1"/>
 <a href="jordan1h.html#K1" target="_self">RealOrd</a>  is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<b>thus </b><a NAME="E4:15_1_1"/>
 <a href="jordan1h.html#K1" target="_self">RealOrd</a>  is <a href="relat_2.html#V1">reflexive</a>
 <b>by </b><i><a class="txt" href="jordan1h.html#E11"><i><font color="Green">Lemma4</font></i></a>, <a class="ref" href="jordan1h.html#T10" target="_self">Th10</a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>

<b>thus </b><a NAME="E5:15_1_1"/>
 <a href="jordan1h.html#K1" target="_self">RealOrd</a>  is <a href="relat_2.html#V4">antisymmetric</a>
 <b>by </b><i><a class="txt" href="jordan1h.html#E12"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="jordan1h.html#T10" target="_self">Th10</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/>

<b>thus </b><a NAME="E6:15_1_1"/>
 <a href="jordan1h.html#K1" target="_self">RealOrd</a>  is <a href="relat_2.html#V8">transitive</a>
 <b>by </b><i><a class="txt" href="jordan1h.html#E13"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="jordan1h.html#T10" target="_self">Th10</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>

<b>thus </b><a NAME="E7:15_1_1"/>
 <a href="jordan1h.html#K1" target="_self">RealOrd</a>  <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> 
 <b>by </b><i><a class="txt" href="jordan1h.html#E11"><i><font color="Green">Lemma4</font></i></a>, <a class="ref" href="jordan1h.html#T10" target="_self">Th10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>,<a class="ref" href="orders_1.html#D5">ORDERS_1:def 5</a><br/>

<b>thus </b><a NAME="E8:15_1_1"/>
 <a href="jordan1h.html#K1" target="_self">RealOrd</a>  <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> 
 <b>by </b><i><a class="txt" href="jordan1h.html#E13"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="jordan1h.html#T10" target="_self">Th10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a><br/>

<b>thus </b><a NAME="E9:15_1_1"/>
 <a href="jordan1h.html#K1" target="_self">RealOrd</a>  <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> 
 <b>by </b><i><a class="txt" href="jordan1h.html#E12"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="jordan1h.html#T10" target="_self">Th10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a><br/>

<b>thus </b><a NAME="E10:15_1_1"/>
 <a href="jordan1h.html#K1" target="_self">RealOrd</a>  <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> 
 <b>by </b><i><a class="txt" href="jordan1h.html#E14"><i><font color="Green">Lemma7</font></i></a>, <a class="ref" href="jordan1h.html#T10" target="_self">Th10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a><br/>


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