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<div class="add">

<b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</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="E1:179"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 holds <br/><font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:179"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:179"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:179"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_1"/>
( <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or ( <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) or ( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_1"><b>suppose </b></a><a NAME="E1:179_1_1"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:179_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:179_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:179_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:179_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">a</font> = <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span>;<br/><a NAME="E3:179_1_2"/>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E4:179_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E5:179_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E6:179_1_2"/>
<font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , </i>;<br/><a NAME="E7:179_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E8:179_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i/>;<br/><a NAME="E9:179_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>;<br/><a NAME="E10:179_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E11:179_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_1_2_1"/>
( <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> or <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_2_1_1"><b>suppose </b></a><a NAME="E1:179_1_2_1_1"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:179_1_2_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:179_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T93">XCMPLX_1:93</a></i>;<br/><b>hence </b><a NAME="E4:179_1_2_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_2_1_2"><b>suppose </b></a><a NAME="E1:179_1_2_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:179_1_2_1_2"/>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E3:179_1_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:179_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:179_1_2_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T93">XCMPLX_1:93</a></i>;<br/><a NAME="E6:179_1_2_1_2"/>
<font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="txt" href="xreal_1.html#E7"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E7:179_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:179_1_2_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:179_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:179_1_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">a</font> = <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span>;<br/><a NAME="E3:179_1_3"/>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E4:179_1_3"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E5:179_1_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E6:179_1_3"/>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , </i>;<br/><a NAME="E7:179_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E8:179_1_3"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="txt" href="xreal_1.html#E11:179_1_3"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E9:179_1_3"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>;<br/><a NAME="E10:179_1_3"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E11:179_1_3"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_1_3_1"/>
( <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> or <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_3_1_1"><b>suppose </b></a><a NAME="E1:179_1_3_1_1"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:179_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T93">XCMPLX_1:93</a></i>;<br/><b>hence </b><a NAME="E3:179_1_3_1_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="xreal_1.html#E11:179_1_3"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_3_1_2"><b>suppose </b></a><a NAME="E1:179_1_3_1_2"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_1_3_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
