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<div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:7"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>and </b><br/><a NAME="E5:7"/><i><font color="Green">E10</font></i>: 
 <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E6:7"/><i><font color="Green">E11</font></i>: 
 <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D6">INT_1:def 6</a></i>;<br/>
<a NAME="E7:7"/><i><font color="Green">E12</font></i>: 
 <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D6">INT_1:def 6</a></i>;<br/>
<a NAME="E8:7"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E9:7"/><b>then </b><i><font color="Green">E13</font></i>: 
( <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="topreala.html#E1:7"><i><font color="Green">E6</font></i></a>, <a class="txt" href="topreala.html#E3:7"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>suppose </b></a><a NAME="E1:7_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:7_1_1"/><b>then </b>
(  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="topreala.html#E1:7"><i><font color="Green">E6</font></i></a>, <a class="txt" href="topreala.html#E4:7"><i><font color="Green">E9</font></i></a>, <a class="ref" href="topreala.html#T3" target="_self">Th3</a>, <a class="ref" href="topreala.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E3:7_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topreala.html#E5:7"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2"><b>suppose </b></a><a NAME="E1:7_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:7_1_2"/><b>then </b>
(  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="topreala.html#E2:7"><i><font color="Green">E7</font></i></a>, <a class="txt" href="topreala.html#E3:7"><i><font color="Green">E8</font></i></a>, <a class="ref" href="topreala.html#T3" target="_self">Th3</a>, <a class="ref" href="topreala.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E3:7_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topreala.html#E5:7"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3"><b>suppose </b></a><a NAME="E1:7_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:7_1_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:7_1_3"/><b>then </b><i><font color="Green">E14</font></i>: 
( <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> )
 <b>by </b><i><a class="txt" href="topreala.html#E9:7"><i><font color="Green">E13</font></i></a>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="int_1.html#K1">/]</a></span>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="topreala.html#E5:7"><i><font color="Green">E10</font></i></a>, <a class="txt" href="topreala.html#E6:7"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topreala.html#E7:7"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topreala.html#E3:7_1_3"><i><font color="Green">E14</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_4"><b>suppose </b></a><a NAME="E1:7_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:7_1_4"/><b>then </b><i><font color="Green">E14</font></i>: 
( <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="txt" href="topreala.html#E2:7"><i><font color="Green">E7</font></i></a>, <a class="txt" href="topreala.html#E4:7"><i><font color="Green">E9</font></i></a>, <a class="ref" href="topreala.html#T5" target="_self">Th5</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="int_1.html#K1">/]</a></span>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="topreala.html#E5:7"><i><font color="Green">E10</font></i></a>, <a class="txt" href="topreala.html#E6:7"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topreala.html#E7:7"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topreala.html#E2:7_1_4"><i><font color="Green">E14</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
