<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:51"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E2:51"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E3:51"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1"><b>case </b></a><a NAME="E1:51_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="txt" href="int_3.html#E1:51_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default">1</span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>
<br/>.= 
0

;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:51_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:51_1_1_1"/><i><font color="Green">E43</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="txt" href="int_3.html#E6:51_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><a NAME="E9:51_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:51_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:51_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="int_3.html#E1:51_1_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:51_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:51_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E5:51_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="txt" href="int_3.html#E3:51"><i><font color="Green">E39</font></i></a>, <a class="txt" href="int_3.html#E1:51_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="int_3.html#E8:51_1_1_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:51_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:51_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:51_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="int_3.html#E3:51"><i><font color="Green">E39</font></i></a>, <a class="txt" href="int_3.html#E1:51_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="int_3.html#E8:51_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:51_1_1_1_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E10:51_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="txt" href="int_3.html#E8:51_1_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><b>hence </b><a NAME="E11:51_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E3:51"><i><font color="Green">E39</font></i></a>, <a class="txt" href="int_3.html#E1:51_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="int_3.html#E3:51_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="int_3.html#E6:51_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="int_3.html#E9:51_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_2"><b>case </b></a><a NAME="E1:51_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E3:51_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:51_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E2:51_1_1_2_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:51_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E1:51_1_1_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:51_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E2:51_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
;<br/><a NAME="E5:51_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> ;<br/>
<a NAME="E1:51_1_1_2_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E2:51_1_1_2_3"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> 
 
<b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E3:51_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:51_1_1_2_3_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:51_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="int_3.html#E2:51_1_1_2_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E2:51_1_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:51_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="int_3.html#E2:51_1_1_2_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<b>hence </b><a NAME="E3:51_1_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E1:51_1_1_2_3_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:51_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="txt" href="int_3.html#E1:51_1_1_2_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:51_1_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:51_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> ;<br/>
<a NAME="E1:51_1_1_2_4"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> 
 
<b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E2:51_1_1_2_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E2:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:51_1_1_2_4"/><b>then </b>
0 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="int_3.html#E1:51_1_1_2_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E4:51_1_1_2_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:51_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="int_3.html#E6:51_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:51_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:51_1_1_2"/><i><font color="Green">E46</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="txt" href="int_3.html#E11:51_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><a NAME="E14:51_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_1_2_5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:51_1_1_2_5"/><i><font color="Green">E48</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<a NAME="E3:51_1_1_2_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:51_1_1_2_5_1"/>
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E1:51_1_1_2_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E2:51_1_1_2_5"><i><font color="Green">E48</font></i></a>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E2:51_1_1_2_5_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:51_1_1_2_5"/><b>then </b>
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:51_1_1_2_5"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:51_1_1_2_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E7:51_1_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="int_3.html#E3:51"><i><font color="Green">E39</font></i></a>, <a class="txt" href="int_3.html#E3:51_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="int_3.html#E13:51_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="int_3.html#E5:51_1_1_2_5"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E15:51_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="txt" href="int_3.html#E3:51"><i><font color="Green">E39</font></i></a>, <a class="txt" href="int_3.html#E4:51_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="int_3.html#E8:51_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="int_3.html#E11:51_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="int_3.html#E13:51_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:51"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
