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<b>let </b><font color="Maroon">a</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">b</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> or ( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> ) or ( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>suppose </b></a><a NAME="E1:2_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:2_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">c</font> = 
0 <a href="binarith.html#K3">-'</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2"><b>suppose </b></a><a NAME="E1:2_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> )
 ;<br/><div class="add"><a NAME="E2:2_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E3:2_1_2"/>
<font color="Maroon">a</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">c</font> = 
0
<b>by </b><i>, <a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_3"><b>suppose </b></a><a NAME="E1:2_1_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> )
 ;<br/><div class="add"><a NAME="E2:2_1_3"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E3:2_1_3"/>
<font color="Maroon">a</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span>

<br/>.= 
<font color="Maroon">a</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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