<?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="yellow11.html#V1">modular</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="lattice8.html#M1" target="_self">ExtensionSeq2</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">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="E1:52"/><i><font color="Green">E41</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="E2:52"/>
<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="E5:52"/><i><font color="Green">E42</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="52_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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="E1:52_1_1"/><i><font color="Green">E43</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="E2:52_1_1"/><i><font color="Green">E44</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="E4:52_1_1"/><i><font color="Green">E45</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="lattice8.html#E1:52_1_1"><i><font color="Green">E43</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="E6:52_1_1"/><i><font color="Green">E46</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="lattice8.html#E2:52_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E7:52_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="E8:52_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="lattice8.html#E4:52_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="lattice8.html#E6:52_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="lattice8.html#T32" target="_self">Th32</a></i>;<br/><b>hence </b><a NAME="E9:52_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="E2:52_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="E6:52"/>
<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="52_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="E1:52_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="E3:52_2"/><i><font color="Green">E43</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="E5:52_2"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E6:52_2"/><i><font color="Green">E44</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E7:52_2"/><i><font color="Green">E45</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="lattice8.html#E3:52_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lattice8.html#E6:52_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E8:52_2"/><i><font color="Green">E46</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="E9:52_2"/><i><font color="Green">E47</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="E10:52_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="lattice8.html#E6:52_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E11:52_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="E12:52_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="lattice8.html#E1:52"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E13:52_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="lattice8.html#E9:52_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E14:52_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="lattice8.html#E7:52_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="lattice8.html#E8:52_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:52"/><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="lattice8.html#E5:52"><i><font color="Green">E42</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="E9:52"/><i><font color="Green">E43</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="E10:52"/>
 <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="E11:52"/><i><font color="Green">E44</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="lattice8.html#E9:52"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E13:52"/><i><font color="Green">E45</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="52_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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="E1:52_3_1"/><i><font color="Green">E46</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="E2:52_3_1"/><i><font color="Green">E47</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="E4:52_3_1"/><i><font color="Green">E48</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="lattice8.html#E1:52_3_1"><i><font color="Green">E46</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="E6:52_3_1"/><i><font color="Green">E49</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="lattice8.html#E2:52_3_1"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E7:52_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="E8:52_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="lattice8.html#E4:52_3_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="lattice8.html#E6:52_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self">Th31</a></i>;<br/><b>hence </b><a NAME="E9:52_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="E2:52_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="E14:52"/><i><font color="Green">E46</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="E15:52"/><i><font color="Green">E47</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="E17:52"/><i><font color="Green">E48</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="E18:52"/><i><font color="Green">E49</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="lattice8.html#E14:52"><i><font color="Green">E46</font></i></a>, <a class="txt" href="lattice8.html#E15:52"><i><font color="Green">E47</font></i></a>);<br/></i>
<a NAME="E19:52"/><i><font color="Green">E50</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="52_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_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="52_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="E1:52_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="E3:52_4_1"/><i><font color="Green">E51</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="lattice8.html#E17:52"><i><font color="Green">E48</font></i></a>, <a class="txt" href="lattice8.html#E3:52_4_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E5:52_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="lattice8.html#E18:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice8.html#E3:52_4_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:52_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="E2:52_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="E4:52_4"/><i><font color="Green">E51</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="E5:52_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="lattice8.html#E18:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice8.html#E4:52_4"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:52_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="lattice8.html#E17:52"><i><font color="Green">E48</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="E20:52"/>
<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="52_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="E1:52_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="lattice8.html#E17:52"><i><font color="Green">E48</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="E4:52_5"/>
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E5:52_5"/><i><font color="Green">E51</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E6:52_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="lattice8.html#E5:52_5"><i><font color="Green">E51</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:52_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="lattice8.html#E18:52"><i><font color="Green">E49</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="E22:52"/><i><font color="Green">E51</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="52_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_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="52_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="E1:52_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="E3:52_6_1"/><i><font color="Green">E52</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="E4:52_6_1"/><i><font color="Green">E53</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="lattice8.html#E3:52_6_1"><i><font color="Green">E52</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="lattice8.html#E17:52"><i><font color="Green">E48</font></i></a>, <a class="txt" href="lattice8.html#E3:52_6_1"><i><font color="Green">E52</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="E7:52_6_1"/>
<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E8:52_6_1"/><i><font color="Green">E54</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E9:52_6_1"/><i><font color="Green">E55</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="lattice8.html#E8:52_6_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E10:52_6_1"/><i><font color="Green">E56</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="lattice8.html#E8:52_6_1"><i><font color="Green">E54</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="lattice8.html#E3:52_6_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="lattice8.html#E4:52_6_1"><i><font color="Green">E53</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="lattice8.html#E18:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice8.html#E9:52_6_1"><i><font color="Green">E55</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="lattice8.html#E10:52_6_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E12:52_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="E2:52_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="E4:52_6"/><i><font color="Green">E52</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="E6:52_6"/>
<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E7:52_6"/><i><font color="Green">E53</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E8:52_6"/><i><font color="Green">E54</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="lattice8.html#E7:52_6"><i><font color="Green">E53</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:52_6"/><i><font color="Green">E55</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="lattice8.html#E7:52_6"><i><font color="Green">E53</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:52_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="E11:52_6"/><b>then </b><i><font color="Green">E56</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="lattice8.html#E17:52"><i><font color="Green">E48</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="lattice8.html#E4:52_6"><i><font color="Green">E52</font></i></a>, <a class="txt" href="lattice8.html#E9:52_6"><i><font color="Green">E55</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="lattice8.html#E18:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice8.html#E8:52_6"><i><font color="Green">E54</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="lattice8.html#E17:52"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/>
<b>hence </b><a NAME="E13:52_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="lattice8.html#E11:52_6"><i><font color="Green">E56</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="E23:52"/>
<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="52_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_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="52_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="E1:52_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="E3:52_7_1"/><i><font color="Green">E52</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="E4:52_7_1"/><i><font color="Green">E53</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="E6:52_7_1"/><i><font color="Green">E54</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="lattice8.html#E4:52_7_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:52_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="lattice8.html#E3:52_7_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="lattice8.html#E6:52_7_1"><i><font color="Green">E54</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="E2:52_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="E4:52_7"/><i><font color="Green">E52</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="E5:52_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="E6:52_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="lattice8.html#E4:52_7"><i><font color="Green">E52</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="lattice8.html#E1:52"><i><font color="Green">E41</font></i></a>, <a class="txt" href="lattice8.html#E13:52"><i><font color="Green">E45</font></i></a>, <a class="txt" href="lattice8.html#E22:52"><i><font color="Green">E51</font></i></a>, <a class="ref" href="lattice5.html#T3">LATTICE5:3</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="lattice8.html#E19:52"><i><font color="Green">E50</font></i></a>, <a class="ref" href="lattice5.html#T1">LATTICE5:1</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="lattice8.html#E9:52"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lattice8.html#E11:52"><i><font color="Green">E44</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="E26:52"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="lattice5.html#V2">zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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">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="E2:52_9"/><i><font color="Green">E53</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="lattice8.html#E1:52"><i><font color="Green">E41</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="E4:52_9"/><i><font color="Green">E54</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="lattice8.html#E2:52_9"><i><font color="Green">E53</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="E6:52_9"/>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E7:52_9"/><i><font color="Green">E55</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E8:52_9"/><i><font color="Green">E56</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="lattice8.html#E7:52_9"><i><font color="Green">E55</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="lattice8.html#E2:52_9"><i><font color="Green">E53</font></i></a>, <a class="txt" href="lattice8.html#E4:52_9"><i><font color="Green">E54</font></i></a>, <a class="txt" href="lattice8.html#E7:52_9"><i><font color="Green">E55</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:52_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="lattice8.html#E8:52_9"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E11:52_9"/><b>then </b><i><font color="Green">E57</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="E12:52_9"/><i><font color="Green">E58</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">.</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="lattice8.html#E11:52_9"><i><font color="Green">E57</font></i></a>, <a class="txt" href="lattice8.html#E12:52_9"><i><font color="Green">E58</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">.</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">LATTICE5:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E27:52"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="lattice5.html#V1">symmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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">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="E2:52_10"/><i><font color="Green">E54</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="lattice8.html#E1:52"><i><font color="Green">E41</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="E4:52_10"/><i><font color="Green">E55</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="lattice8.html#E2:52_10"><i><font color="Green">E54</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="E6:52_10"/>
<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E7:52_10"/><i><font color="Green">E56</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E8:52_10"/><i><font color="Green">E57</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="lattice8.html#E7:52_10"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:52_10"/><i><font color="Green">E58</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="lattice8.html#E7:52_10"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:52_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="lattice8.html#E8:52_10"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E11:52_10"/><b>then </b><i><font color="Green">E59</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="E12:52_10"/><i><font color="Green">E60</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="lattice8.html#E2:52_10"><i><font color="Green">E54</font></i></a>, <a class="txt" href="lattice8.html#E4:52_10"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E7:52_10"><i><font color="Green">E56</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="E14:52_10"/><i><font color="Green">E61</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="lattice8.html#E1:52"><i><font color="Green">E41</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="E16:52_10"/><i><font color="Green">E62</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="lattice8.html#E14:52_10"><i><font color="Green">E61</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="E18:52_10"/>
<font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E19:52_10"/><i><font color="Green">E63</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E20:52_10"/><i><font color="Green">E64</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="lattice8.html#E19:52_10"><i><font color="Green">E63</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E21:52_10"/><i><font color="Green">E65</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="lattice8.html#E19:52_10"><i><font color="Green">E63</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E22:52_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="lattice8.html#E20:52_10"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E23:52_10"/><b>then </b><i><font color="Green">E66</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="E24:52_10"/><i><font color="Green">E67</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="lattice8.html#E14:52_10"><i><font color="Green">E61</font></i></a>, <a class="txt" href="lattice8.html#E16:52_10"><i><font color="Green">E62</font></i></a>, <a class="txt" href="lattice8.html#E19:52_10"><i><font color="Green">E63</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="52_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_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="52_10_1_1"><b>suppose </b></a><a NAME="E1:52_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="E2:52_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="lattice8.html#E9:52_10"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice8.html#E21:52_10"><i><font color="Green">E65</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self">Th31</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="lattice8.html#E12:52_10"><i><font color="Green">E60</font></i></a>, <a class="txt" href="lattice8.html#E14:52_10"><i><font color="Green">E61</font></i></a>, <a class="txt" href="lattice8.html#E16:52_10"><i><font color="Green">E62</font></i></a>, <a class="txt" href="lattice8.html#E19:52_10"><i><font color="Green">E63</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E4:52_10_1_1"/><i><font color="Green">E68</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">.</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="lattice8.html#E23:52_10"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice8.html#E4:52_10_1_1"><i><font color="Green">E68</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">.</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">.</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">LATTICE5:def 6</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="lattice8.html#E23:52_10"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice8.html#E4:52_10_1_1"><i><font color="Green">E68</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">.</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="52_10_1_2"><b>suppose </b></a><a NAME="E1:52_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="E2:52_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="lattice8.html#E9:52_10"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice8.html#E21:52_10"><i><font color="Green">E65</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self">Th31</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="lattice8.html#E2:52_10"><i><font color="Green">E54</font></i></a>, <a class="txt" href="lattice8.html#E4:52_10"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E7:52_10"><i><font color="Green">E56</font></i></a>, <a class="txt" href="lattice8.html#E24:52_10"><i><font color="Green">E67</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E4:52_10_1_2"/><i><font color="Green">E68</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">.</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="lattice8.html#E11:52_10"><i><font color="Green">E59</font></i></a>, <a class="txt" href="lattice8.html#E4:52_10_1_2"><i><font color="Green">E68</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">.</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">.</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">LATTICE5:def 6</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="lattice8.html#E11:52_10"><i><font color="Green">E59</font></i></a>, <a class="txt" href="lattice8.html#E4:52_10_1_2"><i><font color="Green">E68</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">.</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="E28:52"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="lattice5.html#V3">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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">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="E2:52_11"/><i><font color="Green">E54</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="lattice8.html#E1:52"><i><font color="Green">E41</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="E4:52_11"/><i><font color="Green">E55</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="lattice8.html#E2:52_11"><i><font color="Green">E54</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="E6:52_11"/>
<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E7:52_11"/><i><font color="Green">E56</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E8:52_11"/><i><font color="Green">E57</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="lattice8.html#E7:52_11"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:52_11"/><i><font color="Green">E58</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="lattice8.html#E7:52_11"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:52_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="lattice8.html#E8:52_11"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E11:52_11"/><b>then </b><i><font color="Green">E59</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="E12:52_11"/><i><font color="Green">E60</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="lattice8.html#E2:52_11"><i><font color="Green">E54</font></i></a>, <a class="txt" href="lattice8.html#E4:52_11"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E7:52_11"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E13:52_11"/><i><font color="Green">E61</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="E15:52_11"/><i><font color="Green">E62</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="lattice8.html#E1:52"><i><font color="Green">E41</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="E17:52_11"/><i><font color="Green">E63</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="lattice8.html#E15:52_11"><i><font color="Green">E62</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="E19:52_11"/>
<font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E20:52_11"/><i><font color="Green">E64</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E21:52_11"/><i><font color="Green">E65</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="lattice8.html#E20:52_11"><i><font color="Green">E64</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E22:52_11"/><i><font color="Green">E66</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="lattice8.html#E20:52_11"><i><font color="Green">E64</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E23:52_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="lattice8.html#E21:52_11"><i><font color="Green">E65</font></i></a></i>;<br/>
<a NAME="E24:52_11"/><b>then </b><i><font color="Green">E67</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="E25:52_11"/><i><font color="Green">E68</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="lattice8.html#E15:52_11"><i><font color="Green">E62</font></i></a>, <a class="txt" href="lattice8.html#E17:52_11"><i><font color="Green">E63</font></i></a>, <a class="txt" href="lattice8.html#E20:52_11"><i><font color="Green">E64</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E26:52_11"/><i><font color="Green">E69</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="E28:52_11"/><i><font color="Green">E70</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="lattice8.html#E1:52"><i><font color="Green">E41</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="E30:52_11"/><i><font color="Green">E71</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="lattice8.html#E28:52_11"><i><font color="Green">E70</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="E32:52_11"/>
<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E33:52_11"/><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>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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E34:52_11"/><i><font color="Green">E73</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="lattice8.html#E33:52_11"><i><font color="Green">E72</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E35:52_11"/><i><font color="Green">E74</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="lattice8.html#E33:52_11"><i><font color="Green">E72</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E36:52_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="lattice8.html#E34:52_11"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E37:52_11"/><b>then </b><i><font color="Green">E75</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="E38:52_11"/><i><font color="Green">E76</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="lattice8.html#E28:52_11"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice8.html#E30:52_11"><i><font color="Green">E71</font></i></a>, <a class="txt" href="lattice8.html#E33:52_11"><i><font color="Green">E72</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E39:52_11"/><i><font color="Green">E77</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="52_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_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="52_11_1_1"><b>suppose </b></a><a NAME="E1:52_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="E2:52_11_1_1"/><b>then </b><i><font color="Green">E78</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="lattice8.html#E9:52_11"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice8.html#E22:52_11"><i><font color="Green">E66</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self">Th31</a></i>;<br/><b>thus </b><a NAME="E3:52_11_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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">.</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">.</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="52_11_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_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="E1:52_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="52_11_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_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="E2:52_11_1_1_1_1_1"/><b>then </b><i><font color="Green">E79</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="lattice8.html#E22:52_11"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice8.html#E35:52_11"><i><font color="Green">E74</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E3:52_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="lattice8.html#E2:52_11_1_1"><i><font color="Green">E78</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="lattice8.html#E12:52_11"><i><font color="Green">E60</font></i></a>, <a class="txt" href="lattice8.html#E25:52_11"><i><font color="Green">E68</font></i></a>, <a class="txt" href="lattice8.html#E28:52_11"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice8.html#E30:52_11"><i><font color="Green">E71</font></i></a>, <a class="txt" href="lattice8.html#E33:52_11"><i><font color="Green">E72</font></i></a>, <a class="txt" href="lattice8.html#E2:52_11_1_1_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E80</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E37:52_11"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice8.html#E39:52_11"><i><font color="Green">E77</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">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font>

;<br/><i><font color="Green">E81</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E37:52_11"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice8.html#E39:52_11"><i><font color="Green">E77</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">.</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">.</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="lattice8.html#E37:52_11"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice8.html#E39:52_11"><i><font color="Green">E77</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">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><b>hence </b><a NAME="E8:52_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">.</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">.</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">.</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="lattice8.html#E5:52_11_1_1_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="lattice8.html#E6:52_11_1_1_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_11_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_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="E2:52_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="lattice8.html#E22:52_11"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice8.html#E35:52_11"><i><font color="Green">E74</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self">Th31</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="lattice8.html#E12:52_11"><i><font color="Green">E60</font></i></a>, <a class="txt" href="lattice8.html#E15:52_11"><i><font color="Green">E62</font></i></a>, <a class="txt" href="lattice8.html#E17:52_11"><i><font color="Green">E63</font></i></a>, <a class="txt" href="lattice8.html#E20:52_11"><i><font color="Green">E64</font></i></a>, <a class="txt" href="lattice8.html#E38:52_11"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice8.html#E2:52_11_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E79</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E24:52_11"><i><font color="Green">E67</font></i></a>, <a class="txt" href="lattice8.html#E26:52_11"><i><font color="Green">E69</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">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font>

;<br/><i><font color="Green">E80</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E24:52_11"><i><font color="Green">E67</font></i></a>, <a class="txt" href="lattice8.html#E26:52_11"><i><font color="Green">E69</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">.</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">.</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="lattice8.html#E24:52_11"><i><font color="Green">E67</font></i></a>, <a class="txt" href="lattice8.html#E26:52_11"><i><font color="Green">E69</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">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><b>hence </b><a NAME="E7:52_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">.</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">.</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">.</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="lattice8.html#E4:52_11_1_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="lattice8.html#E5:52_11_1_1_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</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="52_11_1_2"><b>suppose </b></a><a NAME="E1:52_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="E2:52_11_1_2"/><b>then </b><i><font color="Green">E78</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="lattice8.html#E9:52_11"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice8.html#E22:52_11"><i><font color="Green">E66</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self">Th31</a></i>;<br/><b>thus </b><a NAME="E3:52_11_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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">.</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">.</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="52_11_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_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="E1:52_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="52_11_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_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="E2:52_11_1_2_1_1_1"/><b>then </b><i><font color="Green">E79</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="lattice8.html#E9:52_11"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice8.html#E35:52_11"><i><font color="Green">E74</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E3:52_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="lattice8.html#E2:52_11_1_2"><i><font color="Green">E78</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="lattice8.html#E12:52_11"><i><font color="Green">E60</font></i></a>, <a class="txt" href="lattice8.html#E25:52_11"><i><font color="Green">E68</font></i></a>, <a class="txt" href="lattice8.html#E28:52_11"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice8.html#E30:52_11"><i><font color="Green">E71</font></i></a>, <a class="txt" href="lattice8.html#E33:52_11"><i><font color="Green">E72</font></i></a>, <a class="txt" href="lattice8.html#E2:52_11_1_2_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E80</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E37:52_11"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice8.html#E39:52_11"><i><font color="Green">E77</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">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font>

;<br/><i><font color="Green">E81</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E37:52_11"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice8.html#E39:52_11"><i><font color="Green">E77</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">.</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">.</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="lattice8.html#E37:52_11"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice8.html#E39:52_11"><i><font color="Green">E77</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">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><b>hence </b><a NAME="E8:52_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">.</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">.</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">.</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="lattice8.html#E5:52_11_1_2_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="lattice8.html#E6:52_11_1_2_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_11_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_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="E2:52_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="lattice8.html#E9:52_11"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice8.html#E35:52_11"><i><font color="Green">E74</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self">Th31</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="lattice8.html#E2:52_11"><i><font color="Green">E54</font></i></a>, <a class="txt" href="lattice8.html#E4:52_11"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E7:52_11"><i><font color="Green">E56</font></i></a>, <a class="txt" href="lattice8.html#E25:52_11"><i><font color="Green">E68</font></i></a>, <a class="txt" href="lattice8.html#E38:52_11"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice8.html#E2:52_11_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E79</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E11:52_11"><i><font color="Green">E59</font></i></a>, <a class="txt" href="lattice8.html#E13:52_11"><i><font color="Green">E61</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">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font>

;<br/><i><font color="Green">E80</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E11:52_11"><i><font color="Green">E59</font></i></a>, <a class="txt" href="lattice8.html#E13:52_11"><i><font color="Green">E61</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">.</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">.</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="lattice8.html#E11:52_11"><i><font color="Green">E59</font></i></a>, <a class="txt" href="lattice8.html#E13:52_11"><i><font color="Green">E61</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">.</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>38</sub></font>

;<br/><b>hence </b><a NAME="E7:52_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">.</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">.</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">.</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="lattice8.html#E4:52_11_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="lattice8.html#E5:52_11_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</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="E29:52"/>
 <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="lattice8.html#E26:52"><i><font color="Green">E52</font></i></a>, <a class="txt" href="lattice8.html#E27:52"><i><font color="Green">E53</font></i></a></i>;<br/>


</div>
