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<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>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E44</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E45</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 3 )
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 ;<br/>

<a NAME="E7:18"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> or <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>case </b></a><a NAME="E1:18_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>hence </b><a NAME="E3:18_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>case </b></a><a NAME="E1:18_1_1_2"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2"/>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E3:18_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 3 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E4:18_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:18_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:18"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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