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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b>
 <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font>
;<br/><b>thus </b><a NAME="E2:14_1_1_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_1"><i><font color="Green">E3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2"/><i><font color="Green">E3</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2"><i><font color="Green">E3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/><a NAME="E5:14_1_1_1_2"/>
 <a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="triang_1.html#K1" target="_self">|_2</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> #) ;<br/><a NAME="E7:14_1_1_1_2"/><i><font color="Green">E4</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="triang_1.html#K1" target="_self">|_2</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/><a NAME="E8:14_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_2.html#R6">is_connected_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14"><i><font color="Green">E2</font></i></a>, <a class="ref" href="orders_1.html#D8">ORDERS_1:def 8</a></i>;<br/><a NAME="E9:14_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V6">connected</a>
 <b>by </b><i><a class="ref" href="orders_1.html#T185">ORDERS_1:185</a></i>;<br/><a NAME="E10:14_1_1_1_2"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R6">is_connected_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E7:14_1_1_1_2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>7</sub></font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:14_1_1_1_2"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> holds <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E13:14_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1_1_1_2_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="triang_1.html#E3:14_1_1_1_2_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:14_1_1_1_2_1"/><i><font color="Green">E8</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E6:14_1_1_1_2_1"/>
 <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T62">ORDERS_2:62</a></i>;<br/>
<a NAME="E7:14_1_1_1_2_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="triang_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:14_1_1_1_2_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E5:14_1_1_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E9:14_1_1_1_2_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:14_1_1_1_2_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="triang_1.html#E3:14_1_1_1_2_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E12:14_1_1_1_2_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E10:14_1_1_1_2_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E13:14_1_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="triang_1.html#E9:14_1_1_1_2_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="triang_1.html#E10:14_1_1_1_2_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>7</sub></font>, <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/><a NAME="E15:14_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_1_1_1_2_2"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E4:14_1_1_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E5:14_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="triang_1.html#E1:14_1_1_1_2_2"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E6:14_1_1_1_2_2"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E10</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E10:14_1_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/><div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_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="E1:14_1_1_1_2_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 </span><b>by </b><i><a class="txt" href="triang_1.html#E2:14_1_1_1_2_2_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:14_1_1_1_2_2_1_1_2_1_1_1"/><b>then </b>
 <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#T64">ORDERS_2:64</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_2_1_1_2_1_1_1"/>
 <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E6:14_1_1_1_2_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="triang_1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:14_1_1_1_2_2_1_1_2_1_1_2"/><b>then </b>
 <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#T64">ORDERS_2:64</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_2_1_1_2_1_1_2"/>
 <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E6:14_1_1_1_2_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="triang_1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_1_1_2_2_1_1_2_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:14_1_1_1_2_2_1_1_2_2"/><b>then </b><i><font color="Green">E13</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E10:14_1_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_2_2_1_1_2_2_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 </span><b>by </b><i><a class="txt" href="triang_1.html#E2:14_1_1_1_2_2_1_1_2_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2_1_1_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:14_1_1_1_2_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#T57">ORDERS_2:57</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_2_1_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="triang_1.html#E3:14_1_1_1_2_2_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="orders_2.html#T62">ORDERS_2:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2_1_1_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:14_1_1_1_2_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#T57">ORDERS_2:57</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_2_1_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="triang_1.html#E3:14_1_1_1_2_2_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="orders_2.html#T62">ORDERS_2:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:14_1_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:14_1_1_1_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E16:14_1_1_1_2"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/><a NAME="E17:14_1_1_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E18:14_1_1_1_2"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_4.html#T78">FINSEQ_4:78</a></i>;<br/><a NAME="E19:14_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K2">"</a>  as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E21:14_1_1_1_2"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E22:14_1_1_1_2"/><b>then </b><i><font color="Green">E10</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="triang_1.html#E18:14_1_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/><b>thus </b><a NAME="E24:14_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="triang_1.html#E21:14_1_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E25:14_1_1_1_2"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>assume </b><a NAME="E1:14_1_1_1_2_3"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:14_1_1_1_2_3"/><i><font color="Green">E13</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#D11">CARD_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> as   <a href="card_1.html#NM1">Cardinal</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> as   <a href="card_1.html#NM1">Cardinal</a> ;<br/>
<a NAME="E5:14_1_1_1_2_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E6:14_1_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="triang_1.html#E1:14_1_1_1_2_3"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E7:14_1_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:14_1_1_1_2_3"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
;<br/>
<a NAME="E9:14_1_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E2:14_1_1_1_2_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="card_1.html#T73">CARD_1:73</a></i>;<br/>
<a NAME="E10:14_1_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finsub_1.html#K2">\</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T4">CARD_2:4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:14_1_1_1_2_3"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finsub_1.html#K2">\</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E13:14_1_1_1_2_3"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font> &amp; not <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2_3"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E15:14_1_1_1_2_3"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; not <font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E13:14_1_1_1_2_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="orders_2.html#T57">ORDERS_2:57</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_2_3_1_1"/>
(  not <font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 </span><b>by </b><i><a class="txt" href="triang_1.html#E15:14_1_1_1_2_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_3_1_1_1"/>
not <font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_2_3_1_1_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_1_2_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E13:14_1_1_1_2_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="orders_2.html#T57">ORDERS_2:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_3_1_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E10:14_1_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="triang_1.html#E2:14_1_1_1_2_3_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/><a NAME="E3:14_1_1_1_2_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E15:14_1_1_1_2_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:14_1_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E13:14_1_1_1_2_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="orders_2.html#T57">ORDERS_2:57</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:14_1_1_1_2_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E26:14_1_1_1_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E27:14_1_1_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="triang_1.html#E22:14_1_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="triang_1.html#E22:14_1_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E30:14_1_1_1_2"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="triang_1.html#E27:14_1_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/><a NAME="E31:14_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E25:14_1_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green">E12</font></i></a></i>;<br/><a NAME="E32:14_1_1_1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E33:14_1_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E34:14_1_1_1_2"/><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E32:14_1_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E35:14_1_1_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E36:14_1_1_1_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="triang_1.html#E30:14_1_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="triang_1.html#E33:14_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="triang_1.html#E34:14_1_1_1_2"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
