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<b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">j</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3" target="_self">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E3:53_1_1"/>
<font color="Maroon">j</font> <a href="binarith.html#K3" target="_self">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="binarith.html#E1:53"><i><font color="Green">E29</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:53_1_1"/>
<font color="Maroon">i</font> <a href="binarith.html#K3" target="_self">-'</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="binarith.html#K3" target="_self">-'</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:53_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E3:53_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3" target="_self">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:53_1_2"/>
<font color="Maroon">j</font> <a href="binarith.html#K3" target="_self">-'</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="binarith.html#E1:53"><i><font color="Green">E29</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:53_1_2"/>
<font color="Maroon">i</font> <a href="binarith.html#K3" target="_self">-'</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="binarith.html#K3" target="_self">-'</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="binarith.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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