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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="lattice5.html#M2" target="_self">ExtensionSeq</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:80"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:80"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E5:80"/><i><font color="Green">E63</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  is <a href="ordinal1.html#V6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:80_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E2:80_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E1:80_1_1"><i><font color="Green">E64</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:80_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E2:80_1_1"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E7:80_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><a NAME="E8:80_1_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E4:80_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice5.html#E6:80_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="lattice5.html#T42" target="_self">Th42</a></i>;<br/><b>hence </b><a NAME="E9:80_1_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:80_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  is <a href="ordinal1.html#V6">c=-linear</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:80"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:80_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:80_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>11</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>12</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>13</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:80_2"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E6:80_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E7:80_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:80_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="lattice5.html#E6:80_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E8:80_2"/><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E9:80_2"/><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:80_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E6:80_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E11:80_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
;<br/>
<a NAME="E12:80_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:80"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E13:80_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E9:80_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E14:80_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E7:80_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice5.html#E8:80_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:80"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:80"><i><font color="Green">E63</font></i></a>, <a class="ref" href="hahnban.html#T13">HAHNBAN:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:80"/><i><font color="Green">E64</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E10:80"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E11:80"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="lattice5.html#E9:80"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E13:80"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_3_1"><b>now </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><b>that </b><br/><a NAME="E1:80_3_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E2:80_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_3_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E1:80_3_1"><i><font color="Green">E67</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:80_3_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E2:80_3_1"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E7:80_3_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><a NAME="E8:80_3_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E4:80_3_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="lattice5.html#E6:80_3_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self">Th41</a></i>;<br/><b>hence </b><a NAME="E9:80_3_1"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:80_3"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="ordinal1.html#V6">c=-linear</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> ;<br/>
<a NAME="E14:80"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
;<br/>
<a NAME="E15:80"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</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>  <a href="numbers.html#K5">NAT</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:80"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E18:80"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</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>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="lattice5.html#E14:80"><i><font color="Green">E67</font></i></a>, <a class="txt" href="lattice5.html#E15:80"><i><font color="Green">E68</font></i></a>);<br/></i>
<a NAME="E19:80"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:80_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</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="E1:80_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:80_4_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</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>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="lattice5.html#E17:80"><i><font color="Green">E69</font></i></a>, <a class="txt" href="lattice5.html#E3:80_4_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E5:80_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E18:80"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice5.html#E3:80_4_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:80_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>11</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="E2:80_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_4"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 ;<br/>
<a NAME="E5:80_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E18:80"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice5.html#E4:80_4"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:80_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E17:80"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:80"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:80_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="lattice5.html#E17:80"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>14</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>15</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>16</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:80_5"/>
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E5:80_5"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E6:80_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:80_5"><i><font color="Green">E72</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:80_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="txt" href="lattice5.html#E18:80"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> ;<br/>
<a NAME="E22:80"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:80_6"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</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="E1:80_6_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:80_6_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:80_6_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:80_6_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="lattice5.html#E17:80"><i><font color="Green">E69</font></i></a>, <a class="txt" href="lattice5.html#E3:80_6_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>18</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>19</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>20</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:80_6_1"/>
<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E8:80_6_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E9:80_6_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:80_6_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E10:80_6_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:80_6_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:80_6_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E4:80_6_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E18:80"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice5.html#E9:80_6_1"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E10:80_6_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E12:80_6_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>14</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="E2:80_6"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_6"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>17</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>18</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>19</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:80_6"/>
<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E7:80_6"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E8:80_6"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:80_6"><i><font color="Green">E74</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:80_6"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:80_6"><i><font color="Green">E74</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:80_6"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E11:80_6"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E17:80"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E4:80_6"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E9:80_6"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E18:80"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice5.html#E8:80_6"><i><font color="Green">E75</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E17:80"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/>
<b>hence </b><a NAME="E13:80_6"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E11:80_6"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:80"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:80_7"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</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="E1:80_7_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:80_7_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E4:80_7_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:80_7_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E4:80_7_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:80_7_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="lattice5.html#E3:80_7_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E6:80_7_1"><i><font color="Green">E75</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>14</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="E2:80_7"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_7"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>
<a NAME="E5:80_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<b>hence </b><a NAME="E6:80_7"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="lattice5.html#E4:80_7"><i><font color="Green">E73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E1:80"><i><font color="Green">E62</font></i></a>, <a class="txt" href="lattice5.html#E13:80"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice5.html#E22:80"><i><font color="Green">E72</font></i></a>, <a class="ref" href="lattice5.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E19:80"><i><font color="Green">E71</font></i></a>, <a class="ref" href="lattice5.html#T1" target="_self">Th1</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E9:80"><i><font color="Green">E64</font></i></a>, <a class="txt" href="lattice5.html#E11:80"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E26:80"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="lattice5.html#V2" target="_self">zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D7" target="_self">LATTICE5:def 7</a><br/>

<b>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="E2:80_9"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:80"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_9"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E2:80_9"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>19</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>20</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>21</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:80_9"/>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E7:80_9"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E8:80_9"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:80_9"><i><font color="Green">E76</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>18</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E2:80_9"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice5.html#E4:80_9"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E7:80_9"><i><font color="Green">E76</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:80_9"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:80_9"><i><font color="Green">E77</font></i></a></i>;<br/>
<a NAME="E11:80_9"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:80_9"/><i><font color="Green">E79</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E11:80_9"><i><font color="Green">E78</font></i></a>, <a class="txt" href="lattice5.html#E12:80_9"><i><font color="Green">E79</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>22</sub></font>

<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="lattice5.html#D7" target="_self">Def7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E27:80"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="lattice5.html#V1" target="_self">symmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6" target="_self">LATTICE5:def 6</a><br/>

<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:80_10"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:80"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_10"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E2:80_10"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>20</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>21</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>22</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:80_10"/>
<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E7:80_10"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E8:80_10"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:80_10"><i><font color="Green">E77</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:80_10"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:80_10"><i><font color="Green">E77</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:80_10"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:80_10"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E11:80_10"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:80_10"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:80_10"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E4:80_10"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice5.html#E7:80_10"><i><font color="Green">E77</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:80_10"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:80"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:80_10"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E14:80_10"><i><font color="Green">E82</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>26</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>27</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>28</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E18:80_10"/>
<font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E19:80_10"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E20:80_10"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E19:80_10"><i><font color="Green">E84</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E21:80_10"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E19:80_10"><i><font color="Green">E84</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E22:80_10"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E20:80_10"><i><font color="Green">E85</font></i></a></i>;<br/>
<a NAME="E23:80_10"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E24:80_10"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E14:80_10"><i><font color="Green">E82</font></i></a>, <a class="txt" href="lattice5.html#E16:80_10"><i><font color="Green">E83</font></i></a>, <a class="txt" href="lattice5.html#E19:80_10"><i><font color="Green">E84</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_10_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> or <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_1"><b>suppose </b></a><a NAME="E1:80_10_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><div class="add"><a NAME="E2:80_10_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E9:80_10"><i><font color="Green">E79</font></i></a>, <a class="txt" href="lattice5.html#E21:80_10"><i><font color="Green">E86</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self">Th41</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>25</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E12:80_10"><i><font color="Green">E81</font></i></a>, <a class="txt" href="lattice5.html#E14:80_10"><i><font color="Green">E82</font></i></a>, <a class="txt" href="lattice5.html#E16:80_10"><i><font color="Green">E83</font></i></a>, <a class="txt" href="lattice5.html#E19:80_10"><i><font color="Green">E84</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E4:80_10_1_1"/><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E23:80_10"><i><font color="Green">E87</font></i></a>, <a class="txt" href="lattice5.html#E4:80_10_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>26</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font>

<br/>.= 
<font color="Maroon">c<sub>26</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="ref" href="lattice5.html#D6" target="_self">Def6</a></i>
<br/>.= 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E23:80_10"><i><font color="Green">E87</font></i></a>, <a class="txt" href="lattice5.html#E4:80_10_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_2"><b>suppose </b></a><a NAME="E1:80_10_1_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><a NAME="E2:80_10_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E9:80_10"><i><font color="Green">E79</font></i></a>, <a class="txt" href="lattice5.html#E21:80_10"><i><font color="Green">E86</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self">Th41</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E2:80_10"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E4:80_10"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice5.html#E7:80_10"><i><font color="Green">E77</font></i></a>, <a class="txt" href="lattice5.html#E24:80_10"><i><font color="Green">E88</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E4:80_10_1_2"/><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E11:80_10"><i><font color="Green">E80</font></i></a>, <a class="txt" href="lattice5.html#E4:80_10_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font>

<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="ref" href="lattice5.html#D6" target="_self">Def6</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E11:80_10"><i><font color="Green">E80</font></i></a>, <a class="txt" href="lattice5.html#E4:80_10_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>

;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E28:80"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="lattice5.html#V3" target="_self">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" target="_self">LATTICE5:def 8</a><br/>

<b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:80_11"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:80"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_11"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E2:80_11"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>21</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>22</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>23</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:80_11"/>
<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E7:80_11"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E8:80_11"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:80_11"><i><font color="Green">E77</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:80_11"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:80_11"><i><font color="Green">E77</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:80_11"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:80_11"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E11:80_11"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:80_11"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:80_11"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E4:80_11"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice5.html#E7:80_11"><i><font color="Green">E77</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E13:80_11"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:80_11"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:80"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:80_11"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E15:80_11"><i><font color="Green">E83</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>27</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>28</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>29</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E19:80_11"/>
<font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E20:80_11"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E21:80_11"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E20:80_11"><i><font color="Green">E85</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E22:80_11"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E20:80_11"><i><font color="Green">E85</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E23:80_11"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E21:80_11"><i><font color="Green">E86</font></i></a></i>;<br/>
<a NAME="E24:80_11"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E25:80_11"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E15:80_11"><i><font color="Green">E83</font></i></a>, <a class="txt" href="lattice5.html#E17:80_11"><i><font color="Green">E84</font></i></a>, <a class="txt" href="lattice5.html#E20:80_11"><i><font color="Green">E85</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E26:80_11"/><i><font color="Green">E90</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>30</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:80_11"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:80"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E30:80_11"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E28:80_11"><i><font color="Green">E91</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>32</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>33</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>34</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>35</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E32:80_11"/>
<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E33:80_11"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E34:80_11"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E33:80_11"><i><font color="Green">E93</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E35:80_11"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E33:80_11"><i><font color="Green">E93</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E36:80_11"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E34:80_11"><i><font color="Green">E94</font></i></a></i>;<br/>
<a NAME="E37:80_11"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E38:80_11"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E28:80_11"><i><font color="Green">E91</font></i></a>, <a class="txt" href="lattice5.html#E30:80_11"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E33:80_11"><i><font color="Green">E93</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E39:80_11"/><i><font color="Green">E98</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>32</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_11_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> or <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_11_1_1"><b>suppose </b></a><a NAME="E1:80_11_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/><div class="add"><a NAME="E2:80_11_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E9:80_11"><i><font color="Green">E79</font></i></a>, <a class="txt" href="lattice5.html#E22:80_11"><i><font color="Green">E87</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self">Th41</a></i>;<br/><b>thus </b><a NAME="E3:80_11_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_11_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_11_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="E1:80_11_1_1_1_1"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> or <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_11_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:80_11_1_1_1_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/><div class="add"><a NAME="E2:80_11_1_1_1_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E22:80_11"><i><font color="Green">E87</font></i></a>, <a class="txt" href="lattice5.html#E35:80_11"><i><font color="Green">E95</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E3:80_11_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:80_11_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>32</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E12:80_11"><i><font color="Green">E81</font></i></a>, <a class="txt" href="lattice5.html#E25:80_11"><i><font color="Green">E89</font></i></a>, <a class="txt" href="lattice5.html#E28:80_11"><i><font color="Green">E91</font></i></a>, <a class="txt" href="lattice5.html#E30:80_11"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E33:80_11"><i><font color="Green">E93</font></i></a>, <a class="txt" href="lattice5.html#E2:80_11_1_1_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E101</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:80_11"><i><font color="Green">E96</font></i></a>, <a class="txt" href="lattice5.html#E39:80_11"><i><font color="Green">E98</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font>

;<br/><i><font color="Green">E102</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:80_11"><i><font color="Green">E96</font></i></a>, <a class="txt" href="lattice5.html#E39:80_11"><i><font color="Green">E98</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:80_11"><i><font color="Green">E96</font></i></a>, <a class="txt" href="lattice5.html#E39:80_11"><i><font color="Green">E98</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><b>hence </b><a NAME="E8:80_11_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:80_11_1_1_1_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="lattice5.html#E6:80_11_1_1_1_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_11_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:80_11_1_1_1_1_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/><div class="add"><a NAME="E2:80_11_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E22:80_11"><i><font color="Green">E87</font></i></a>, <a class="txt" href="lattice5.html#E35:80_11"><i><font color="Green">E95</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self">Th41</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>26</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E12:80_11"><i><font color="Green">E81</font></i></a>, <a class="txt" href="lattice5.html#E15:80_11"><i><font color="Green">E83</font></i></a>, <a class="txt" href="lattice5.html#E17:80_11"><i><font color="Green">E84</font></i></a>, <a class="txt" href="lattice5.html#E20:80_11"><i><font color="Green">E85</font></i></a>, <a class="txt" href="lattice5.html#E38:80_11"><i><font color="Green">E97</font></i></a>, <a class="txt" href="lattice5.html#E2:80_11_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E100</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E24:80_11"><i><font color="Green">E88</font></i></a>, <a class="txt" href="lattice5.html#E26:80_11"><i><font color="Green">E90</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>27</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font>

;<br/><i><font color="Green">E101</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E24:80_11"><i><font color="Green">E88</font></i></a>, <a class="txt" href="lattice5.html#E26:80_11"><i><font color="Green">E90</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>27</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E24:80_11"><i><font color="Green">E88</font></i></a>, <a class="txt" href="lattice5.html#E26:80_11"><i><font color="Green">E90</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>27</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><b>hence </b><a NAME="E7:80_11_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:80_11_1_1_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="lattice5.html#E5:80_11_1_1_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_11_1_2"><b>suppose </b></a><a NAME="E1:80_11_1_2"/>
<font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><div class="add"><a NAME="E2:80_11_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E9:80_11"><i><font color="Green">E79</font></i></a>, <a class="txt" href="lattice5.html#E22:80_11"><i><font color="Green">E87</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self">Th41</a></i>;<br/><b>thus </b><a NAME="E3:80_11_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_11_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_11_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="E1:80_11_1_2_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> or <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_11_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:80_11_1_2_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/><div class="add"><a NAME="E2:80_11_1_2_1_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E9:80_11"><i><font color="Green">E79</font></i></a>, <a class="txt" href="lattice5.html#E35:80_11"><i><font color="Green">E95</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E3:80_11_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:80_11_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>32</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E12:80_11"><i><font color="Green">E81</font></i></a>, <a class="txt" href="lattice5.html#E25:80_11"><i><font color="Green">E89</font></i></a>, <a class="txt" href="lattice5.html#E28:80_11"><i><font color="Green">E91</font></i></a>, <a class="txt" href="lattice5.html#E30:80_11"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E33:80_11"><i><font color="Green">E93</font></i></a>, <a class="txt" href="lattice5.html#E2:80_11_1_2_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E101</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:80_11"><i><font color="Green">E96</font></i></a>, <a class="txt" href="lattice5.html#E39:80_11"><i><font color="Green">E98</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font>

;<br/><i><font color="Green">E102</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:80_11"><i><font color="Green">E96</font></i></a>, <a class="txt" href="lattice5.html#E39:80_11"><i><font color="Green">E98</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:80_11"><i><font color="Green">E96</font></i></a>, <a class="txt" href="lattice5.html#E39:80_11"><i><font color="Green">E98</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>33</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><b>hence </b><a NAME="E8:80_11_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:80_11_1_2_1_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="lattice5.html#E6:80_11_1_2_1_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_11_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:80_11_1_2_1_1_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><div class="add"><a NAME="E2:80_11_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E9:80_11"><i><font color="Green">E79</font></i></a>, <a class="txt" href="lattice5.html#E35:80_11"><i><font color="Green">E95</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self">Th41</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>20</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E2:80_11"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E4:80_11"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice5.html#E7:80_11"><i><font color="Green">E77</font></i></a>, <a class="txt" href="lattice5.html#E25:80_11"><i><font color="Green">E89</font></i></a>, <a class="txt" href="lattice5.html#E38:80_11"><i><font color="Green">E97</font></i></a>, <a class="txt" href="lattice5.html#E2:80_11_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E100</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E11:80_11"><i><font color="Green">E80</font></i></a>, <a class="txt" href="lattice5.html#E13:80_11"><i><font color="Green">E82</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font>

;<br/><i><font color="Green">E101</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E11:80_11"><i><font color="Green">E80</font></i></a>, <a class="txt" href="lattice5.html#E13:80_11"><i><font color="Green">E82</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E11:80_11"><i><font color="Green">E80</font></i></a>, <a class="txt" href="lattice5.html#E13:80_11"><i><font color="Green">E82</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><b>hence </b><a NAME="E7:80_11_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:80_11_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="lattice5.html#E5:80_11_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E29:80"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  is   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E26:80"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E27:80"><i><font color="Green">E74</font></i></a></i>;<br/>


</div>
