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

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





<b>assume </b><a NAME="E1:81"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">FS</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">FD</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E2:81"/>
<span class="p1">(<span class="default"><font color="Maroon">S</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">FS</font> = <font color="Maroon">FS</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E5:81"/><i><font color="Green">E37</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  is <a href="ordinal1.html#V6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:81_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 <b>and </b><br/><a NAME="E2:81_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:81_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E17:81"><i><font color="Green">E64</font></i></a></i>;<br/><b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:81_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E18:81"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E7:81_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> or <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E8:81_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E19:81"><i><font color="Green">E66</font></i></a>, , <a class="txt" href="lattice5.html#E9:81"><i><font color="Green">E55</font></i></a></i>;<br/><b>hence </b><a NAME="E9:81_1_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</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="E2:81_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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="E6:81"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">FS</font>,<font color="Maroon">FS</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:81_2"/>
<font color="Maroon">z</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:81_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 ;<br/>
<b>consider </b><font color="Maroon">A'</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d'</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A'</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:81_2"/>
<font color="Maroon">Aq</font>,<font color="Maroon">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A'</font>,<font color="Maroon">d'</font>
 <b>and </b><br/><a NAME="E6:81_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A'</font>,<font color="Maroon">d'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</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">Aq</font>,<font color="Maroon">dq</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="lattice5.html#E8:81_2"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E7:81_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E17:81"><i><font color="Green">E64</font></i></a>, <a class="txt" href="lattice5.html#E18:81"><i><font color="Green">E65</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E8:81_2"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">d'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E9:81_2"/><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A'</font>,<font color="Maroon">A'</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="E10:81_2"/>
<font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E18:81"><i><font color="Green">E65</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E11:81_2"/><b>then </b>
<font color="Maroon">A'</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E12:81_2"/><b>then </b>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E14:81"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E13:81_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d'</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">FS</font>,<font color="Maroon">FS</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E14:81_2"/>
<font color="Maroon">z</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">FS</font>,<font color="Maroon">FS</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E19:81"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:81"/><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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">FS</font>,<font color="Maroon">FS</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E15:81"><i><font color="Green">E63</font></i></a>, <a class="ref" href="hahnban.html#T13">HAHNBAN:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:81"/><i><font color="Green">E55</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E10:81"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">FS</font>,<font color="Maroon">FS</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E11:81"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">FD</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="lattice5.html#E17:81"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E13:81"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">X</font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:81_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:81_3_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:81_3_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:81_3_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/><a NAME="E7:81_3_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> or <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E8:81_3_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E9:81_3_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</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="E2:81_3"/>
<font color="Maroon">X</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> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</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="E14:81"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E15:81"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:81"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E18:81"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E19:81"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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="81_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:81_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:81_4_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="lattice5.html#E3:81_4_1"><i><font color="Green">E69</font></i></a>, </i>;<br/>
<a NAME="E5:81_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:81_4"><i><font color="Green">E70</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:81_4_1"/>
<font color="Maroon">x</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:81_4"/>
<font color="Maroon">x</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:81_4"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 ;<br/>
<a NAME="E5:81_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:81_4"><i><font color="Green">E70</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:81_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:81"/>
<font color="Maroon">F</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:81_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A1</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq1</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:81_5"/>
<font color="Maroon">Aq1</font>,<font color="Maroon">dq1</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A1</font>,<font color="Maroon">d1</font>
 <b>and </b><br/><a NAME="E5:81_5"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">d1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</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">Aq1</font>,<font color="Maroon">dq1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E6:81_5"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:81_5"><i><font color="Green">E72</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:81_5"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</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">LL</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">I</font>,<font color="Olive">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">I</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">PP</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E22:81"/><i><font color="Green">E73</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">F</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_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">F</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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="81_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:81_6_1"/>
<font color="Maroon">x</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">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:81_6_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">j</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">F</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:81_6_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E22:81"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="lattice5.html#E22:81"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<b>consider </b><font color="Maroon">A1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A1</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq1</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:81_6_1"/>
<font color="Maroon">Aq1</font>,<font color="Maroon">dq1</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A1</font>,<font color="Maroon">d1</font>
 <b>and </b><br/><a NAME="E8:81_6_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">d1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</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">Aq1</font>,<font color="Maroon">dq1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E9:81_6_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:81_6_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E10:81_6_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:81_6_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<font color="Maroon">x</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E22:81"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E3:81_6_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d1</font>
<b>by </b><i>, <a class="txt" href="lattice5.html#E8:81_6_1"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E9:81_6_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E12:81_6_1"/>
<font color="Maroon">x</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:81_6"/>
<font color="Maroon">x</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:81_6"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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">A1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A1</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq1</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:81_6"/>
<font color="Maroon">Aq1</font>,<font color="Maroon">dq1</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A1</font>,<font color="Maroon">d1</font>
 <b>and </b><br/><a NAME="E7:81_6"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">d1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</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">Aq1</font>,<font color="Maroon">dq1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E8:81_6"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:81_6"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:81_6"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E11:81_6"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">j</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">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<font color="Maroon">x</font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d1</font>
<b>by </b><i><a class="txt" href="lattice5.html#E22:81"><i><font color="Green">E73</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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/>
<b>hence </b><a NAME="E13:81_6"/>
<font color="Maroon">x</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">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:81"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">I</font>,<font color="Olive">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">I</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_7"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">I</font>,<font color="Olive">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">I</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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="81_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:81_7_1"/>
<font color="Maroon">x</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">I</font>,<font color="Olive">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">I</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">J</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:81_7_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E4:81_7_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:81_7_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:81_7_1"/>
<font color="Maroon">x</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="lattice5.html#E22:81"><i><font color="Green">E73</font></i></a>, </i>;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:81_7"/>
<font color="Maroon">x</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:81_7"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">x</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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="E5:81_7"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<b>hence </b><a NAME="E6:81_7"/>
<font color="Maroon">x</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">I</font>,<font color="Olive">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">I</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="lattice5.html#E22:81"><i><font color="Green">E73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">FS</font>,<font color="Maroon">FS</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">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E14:81"><i><font color="Green">E62</font></i></a>, <a class="txt" href="lattice5.html#E19:81"><i><font color="Green">E66</font></i></a>, , </i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">FD</font>
<b>by </b><i>, </i>
;<br/>
<b>then reconsider </b><font color="Maroon">FD</font> = <font color="Maroon">FD</font> as   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">FS</font>,<font color="Maroon">L</font> <b>by </b><i><a class="txt" href="lattice5.html#E17:81"><i><font color="Green">E64</font></i></a>, <a class="txt" href="lattice5.html#E18:81"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E26:81"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">FD</font> is <a href="lattice5.html#V2" target="_self">zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_9"><b>proof </b></a><div class="add">

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

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:81_9"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E14:81"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:81_9"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A1</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq1</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:81_9"/>
<font color="Maroon">Aq1</font>,<font color="Maroon">dq1</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A1</font>,<font color="Maroon">d1</font>
 <b>and </b><br/><a NAME="E7:81_9"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">d1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</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">Aq1</font>,<font color="Maroon">dq1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E8:81_9"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A1</font> <b>by </b><i>, , , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:81_9"/>
<font color="Maroon">d1</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:81_9"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">d1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:81_9"/><i><font color="Green">E100</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">A1</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">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> = 
<font color="Maroon">d1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">x'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d1</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">x'</font>

<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E27:81"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">FD</font> is <a href="lattice5.html#V1" target="_self">symmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_10"><b>proof </b></a><div class="add">

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

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:81_10"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E14:81"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:81_10"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A1</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq1</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:81_10"/>
<font color="Maroon">Aq1</font>,<font color="Maroon">dq1</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A1</font>,<font color="Maroon">d1</font>
 <b>and </b><br/><a NAME="E7:81_10"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">d1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">Aq1</font>,<font color="Maroon">dq1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E8:81_10"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:81_10"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:81_10"/>
<font color="Maroon">d1</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:81_10"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">d1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:81_10"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:81_10"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E14:81"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:81_10"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A2</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d2</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A2</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq2</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq2</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E18:81_10"/>
<font color="Maroon">Aq2</font>,<font color="Maroon">dq2</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A2</font>,<font color="Maroon">d2</font>
 <b>and </b><br/><a NAME="E19:81_10"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A2</font>,<font color="Maroon">d2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</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">Aq2</font>,<font color="Maroon">dq2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E20:81_10"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E21:81_10"/><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E22:81_10"/>
<font color="Maroon">d2</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E23:81_10"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">d2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E24:81_10"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, , , <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="81_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_10_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> or <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_10_1_1"><b>suppose </b></a><a NAME="E1:81_10_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><a NAME="E2:81_10_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A2</font> <b>by </b><i>, , , , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E4:81_10_1_1"/><i><font color="Green">E119</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A2</font>,<font color="Maroon">A2</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">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">d2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d2</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>

<br/>.= 
<font color="Maroon">d2</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">x'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">x'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_10_1_2"><b>suppose </b></a><a NAME="E1:81_10_1_2"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:81_10_1_2"/><b>then </b>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A1</font> <b>by </b><i>, , , , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E4:81_10_1_2"/><i><font color="Green">E120</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">A1</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">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">d1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d1</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>

<br/>.= 
<font color="Maroon">d1</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">x'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">x'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>

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

</div><b>end;</b></div>
<a NAME="E28:81"/>
<font color="Maroon">FD</font> is <a href="lattice5.html#V3" target="_self">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" target="_self">LATTICE5:def 8</a><br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:81_11"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E14:81"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:81_11"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A1</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq1</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:81_11"/>
<font color="Maroon">Aq1</font>,<font color="Maroon">dq1</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A1</font>,<font color="Maroon">d1</font>
 <b>and </b><br/><a NAME="E7:81_11"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">d1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">Aq1</font>,<font color="Maroon">dq1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E8:81_11"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:81_11"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:81_11"/>
<font color="Maroon">d1</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:81_11"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">d1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:81_11"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E13:81_11"/><i><font color="Green">E131</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">A1</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">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:81_11"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E14:81"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:81_11"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A2</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d2</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A2</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq2</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq2</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E19:81_11"/>
<font color="Maroon">Aq2</font>,<font color="Maroon">dq2</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A2</font>,<font color="Maroon">d2</font>
 <b>and </b><br/><a NAME="E20:81_11"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A2</font>,<font color="Maroon">d2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</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">Aq2</font>,<font color="Maroon">dq2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E21:81_11"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E22:81_11"/><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E23:81_11"/>
<font color="Maroon">d2</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E24:81_11"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">d2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E25:81_11"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E26:81_11"/><i><font color="Green">E139</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A2</font>,<font color="Maroon">A2</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">z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:81_11"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E14:81"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E30:81_11"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A3</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d3</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A3</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq3</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq3</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq3</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E32:81_11"/>
<font color="Maroon">Aq3</font>,<font color="Maroon">dq3</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A3</font>,<font color="Maroon">d3</font>
 <b>and </b><br/><a NAME="E33:81_11"/><i><font color="Green">E224</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A3</font>,<font color="Maroon">d3</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</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">Aq3</font>,<font color="Maroon">dq3</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E34:81_11"/><i><font color="Green">E225</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d3</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E26:81"><i><font color="Green">E93</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E35:81_11"/><i><font color="Green">E226</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A3</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E26:81"><i><font color="Green">E93</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E36:81_11"/>
<font color="Maroon">d3</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E37:81_11"/><b>then </b><i><font color="Green">E227</font></i>: 
<font color="Maroon">d3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E38:81_11"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A3</font>
 
<b>by </b><i>, , <a class="txt" href="lattice5.html#E26:81"><i><font color="Green">E93</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E39:81_11"/><i><font color="Green">E229</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A3</font>,<font color="Maroon">A3</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="81_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_11_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> or <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_1"><b>suppose </b></a><a NAME="E1:81_11_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><a NAME="E2:81_11_1_1"/><b>then </b><i><font color="Green">E230</font></i>: 
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E3:81_11_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_11_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_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="E1:81_11_1_1_1_1"/>
( <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:81_11_1_1_1_1_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:81_11_1_1_1_1_1"/><b>then </b><i><font color="Green">E231</font></i>: 
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A3</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:81_11_1_1_1_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A3</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A3</font> <b>by </b><i>, , , , <a class="txt" href="lattice5.html#E26:81"><i><font color="Green">E93</font></i></a>, , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E232</font></i>: <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">d3</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d3</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>

;<br/><i><font color="Green">E233</font></i>: <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> = 
<font color="Maroon">d3</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">z'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d3</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font>

;<br/><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> = 
<font color="Maroon">d3</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">z'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d3</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>

;<br/><b>hence </b><a NAME="E8:81_11_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E27:81"><i><font color="Green">E101</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:81_11_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><a NAME="E2:81_11_1_1_1_1_2"/><b>then </b>
<font color="Maroon">A3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A2</font> <b>by </b><i>, , , , , , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E234</font></i>: <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">d2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d2</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>

;<br/><i><font color="Green">E235</font></i>: <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> = 
<font color="Maroon">d2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">z'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d2</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font>

;<br/><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> = 
<font color="Maroon">d2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">z'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d2</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>

;<br/><b>hence </b><a NAME="E7:81_11_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E27:81"><i><font color="Green">E101</font></i></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="81_11_1_2"><b>suppose </b></a><a NAME="E1:81_11_1_2"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:81_11_1_2"/><b>then </b><i><font color="Green">E236</font></i>: 
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E3:81_11_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_11_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_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="E1:81_11_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:81_11_1_2_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:81_11_1_2_1_1_1"/><b>then </b><i><font color="Green">E237</font></i>: 
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A3</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:81_11_1_2_1_1_1"/><b>then </b>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A3</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A3</font> <b>by </b><i>, , , , <a class="txt" href="lattice5.html#E26:81"><i><font color="Green">E93</font></i></a>, , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E238</font></i>: <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">d3</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d3</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>

;<br/><i><font color="Green">E239</font></i>: <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> = 
<font color="Maroon">d3</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">z'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d3</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font>

;<br/><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> = 
<font color="Maroon">d3</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">z'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d3</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>

;<br/><b>hence </b><a NAME="E8:81_11_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E27:81"><i><font color="Green">E101</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:81_11_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:81_11_1_2_1_1_2"/><b>then </b>
<font color="Maroon">A3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A1</font> <b>by </b><i>, , , , , , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><i><font color="Green">E240</font></i>: <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">d1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d1</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>

;<br/><i><font color="Green">E241</font></i>: <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> = 
<font color="Maroon">d1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">z'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d1</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font>

;<br/><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> = 
<font color="Maroon">d1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">z'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d1</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>

;<br/><b>hence </b><a NAME="E7:81_11_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E27:81"><i><font color="Green">E101</font></i></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="E29:81"/>
 <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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  is   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">FS</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E22:81"><i><font color="Green">E73</font></i></a>, </i>;<br/>


</div>
