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<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:11_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:11_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">l</font> )
 <b>by </b><i/>;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 ;<br/><a NAME="E2:11_1_1_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font>
 ;<br/><b>hence </b><a NAME="E3:11_1_1_1"/>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> is   <a href="int_1.html#NM1">Integer</a>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">l</font> )
 ;<br/><div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_2_1"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><a NAME="E2:11_1_1_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11_1_1_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E5:11_1_1_2_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:11_1_1_2_1"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> is   <a href="int_1.html#NM1">Integer</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:11_1_1_2_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11_1_1_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>thus </b><a NAME="E5:11_1_1_2_2"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> is   <a href="int_1.html#NM1">Integer</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:11_1_1_2"/>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> is   <a href="int_1.html#NM1">Integer</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 ;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_3_1"/>
<font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><a NAME="E2:11_1_1_3_1"/><b>then </b>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11_1_1_3_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E5:11_1_1_3_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:11_1_1_3_1"/>
<font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> is   <a href="int_1.html#NM1">Integer</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_3_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:11_1_1_3_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11_1_1_3_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>thus </b><a NAME="E5:11_1_1_3_2"/>
<font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> is   <a href="int_1.html#NM1">Integer</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:11_1_1_3"/>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> is   <a href="int_1.html#NM1">Integer</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_4"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">l</font> )
 ;<br/><a NAME="E2:11_1_1_4"/><b>then </b>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E3:11_1_1_4"/>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> is   <a href="int_1.html#NM1">Integer</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:11_1_1"/>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> is <a href="int_1.html#V1" target="_self">integer</a>
 <b>by </b><i>, , , , </i>;<br/>


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