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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be    <a href="lattice8.html#M1" title="LATTICE8:mode.1">ExtensionSeq2</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>, <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">FS</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c3">FS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon" title="c4">FD</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E2:51"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">FS</font> = <font color="Maroon" title="c3">FS</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E5:51"/><i><font color="Green" title="E41">A2</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:51_1_1"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 <b>and </b><br/><a NAME="E2:51_1_1"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon" title="c10">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:51_1_1"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E1:51_1_1"><i><font color="Green" title="E42">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:51_1_1"/><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E2:51_1_1"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/><a NAME="E7:51_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">l</font> or <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> )
 ;<br/><a NAME="E8:51_1_1"/><b>then </b>
( <font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">y</font> or <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E4:51_1_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="lattice8.html#E6:51_1_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="lattice8.html#T32" target="_self" title="LATTICE8:th.32">Th32</a></i>;<br/><b>hence </b><a NAME="E9:51_1_1"/>
<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:51"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">FS</font>,<font color="Maroon" title="c7">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:51_2"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">A'</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c11">d'</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c12">Aq</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c13">dq</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c12">Aq</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:51_2"/>
<font color="Maroon" title="c12">Aq</font>,<font color="Maroon" title="c13">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">d'</font>
 <b>and </b><br/><a NAME="E6:51_2"/><i><font color="Green" title="E43">A8</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">Aq</font>,<font color="Maroon" title="c13">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E7:51_2"/><i><font color="Green" title="E44">A9</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">d'</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:51_2"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="lattice8.html#E6:51_2"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E8:51_2"/><i><font color="Green" title="E45">A10</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">d'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
;<br/>
<a NAME="E9:51_2"/><i><font color="Green" title="E46">A11</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c10">A'</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:51_2"/>
<font color="Maroon" title="c10">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E6:51_2"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E11:51_2"/><b>then </b>
<font color="Maroon" title="c10">A'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E12:51_2"/><b>then </b>
<font color="Maroon" title="c10">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">FS</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:51"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E13:51_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">d'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">FS</font>,<font color="Maroon" title="c7">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E9:51_2"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E14:51_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">FS</font>,<font color="Maroon" title="c7">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:51_2"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="lattice8.html#E8:51_2"><i><font color="Green" title="E45">A10</font></i></a>, <a class="ref" href="partfun1.html#D5" title="PARTFUN1:def.5">PARTFUN1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:51"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">FS</font>,<font color="Maroon" title="c7">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E5:51"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="hahnban.html#T13" title="HAHNBAN:th.13">HAHNBAN:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:51"/><i><font color="Green" title="E42">A12</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 <b>and </b><br/><a NAME="E10:51"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">FS</font>,<font color="Maroon" title="c7">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E11:51"/><i><font color="Green" title="E43">A13</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D5" title="PARTFUN1:def.5">PARTFUN1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">FD</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="lattice8.html#E9:51"><i><font color="Green" title="E42">A12</font></i></a></i>;<br/>
<a NAME="E13:51"/><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon" title="c6">X</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:51_3_1"/><i><font color="Green" title="E45">A15</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <b>and </b><br/><a NAME="E2:51_3_1"/><i><font color="Green" title="E46">A16</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/><b>consider </b><font color="Maroon" title="c12">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:51_3_1"/><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E1:51_3_1"><i><font color="Green" title="E45">A15</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:51_3_1"/><i><font color="Green" title="E48">A18</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E2:51_3_1"><i><font color="Green" title="E46">A16</font></i></a></i>;<br/><a NAME="E7:51_3_1"/>
( <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">l</font> or <font color="Maroon" title="c13">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> )
 ;<br/><a NAME="E8:51_3_1"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">y</font> or <font color="Maroon" title="c11">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E4:51_3_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="lattice8.html#E6:51_3_1"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self" title="LATTICE8:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E9:51_3_1"/>
<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_3"/>
<font color="Maroon" title="c6">X</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">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" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ;<br/>
<a NAME="E14:51"/><i><font color="Green" title="E45">A19</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
;<br/>
<a NAME="E15:51"/><i><font color="Green" title="E46">A20</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c10">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E17:51"/><i><font color="Green" title="E47">A21</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E18:51"/><i><font color="Green" title="E48">A22</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="lattice8.html#E14:51"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="lattice8.html#E15:51"><i><font color="Green" title="E46">A20</font></i></a>);<br/></i>
<a NAME="E19:51"/><i><font color="Green" title="E49">A23</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:51_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_4_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51_4_1"/><i><font color="Green" title="E50">A24</font></i>: 
( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">F</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c12">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lattice8.html#E17:51"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="lattice8.html#E3:51_4_1"><i><font color="Green" title="E50">A24</font></i></a></i>;<br/>
<a NAME="E5:51_4_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E18:51"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="lattice8.html#E3:51_4_1"><i><font color="Green" title="E50">A24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:51_4_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:51_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:51_4"/><i><font color="Green" title="E50">A25</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ;<br/>
<a NAME="E5:51_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E18:51"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="lattice8.html#E4:51_4"><i><font color="Green" title="E50">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:51_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E17:51"><i><font color="Green" title="E47">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:51"/>
<font color="Maroon" title="c10">F</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:51_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">j</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lattice8.html#E17:51"><i><font color="Green" title="E47">A21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">A1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c14">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c13">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c15">Aq1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c16">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c15">Aq1</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:51_5"/>
<font color="Maroon" title="c15">Aq1</font>,<font color="Maroon" title="c16">dq1</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c13">A1</font>,<font color="Maroon" title="c14">d1</font>
 <b>and </b><br/><a NAME="E5:51_5"/><i><font color="Green" title="E50">A26</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">A1</font>,<font color="Maroon" title="c14">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">Aq1</font>,<font color="Maroon" title="c16">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E6:51_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">d1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E5:51_5"><i><font color="Green" title="E50">A26</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:51_5"/>
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="lattice8.html#E18:51"><i><font color="Green" title="E48">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c10">F</font> as  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>set </b><font color="Maroon" title="c12">LL</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c13">PP</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E22:51"/><i><font color="Green" title="E50">A27</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:51_6"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_6_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c11">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51_6_1"/><i><font color="Green" title="E51">A28</font></i>: 
( <font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c11">F</font></span>)</span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:51_6_1"/><i><font color="Green" title="E52">A29</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">F</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:51_6_1"><i><font color="Green" title="E51">A28</font></i></a>, <a class="ref" href="funct_6.html#T89" title="FUNCT_6:th.89">FUNCT_6:89</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">j</font> = <font color="Maroon" title="c15">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lattice8.html#E17:51"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="lattice8.html#E3:51_6_1"><i><font color="Green" title="E51">A28</font></i></a>, <a class="ref" href="funct_6.html#T89" title="FUNCT_6:th.89">FUNCT_6:89</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">A1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c18">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c19">Aq1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c20">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c19">Aq1</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E7:51_6_1"/>
<font color="Maroon" title="c19">Aq1</font>,<font color="Maroon" title="c20">dq1</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c18">d1</font>
 <b>and </b><br/><a NAME="E8:51_6_1"/><i><font color="Green" title="E53">A30</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c18">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">Aq1</font>,<font color="Maroon" title="c20">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E9:51_6_1"/><i><font color="Green" title="E54">A31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E8:51_6_1"><i><font color="Green" title="E53">A30</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E10:51_6_1"/><i><font color="Green" title="E55">A32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">A1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E8:51_6_1"><i><font color="Green" title="E53">A30</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<font color="Maroon" title="c14">x</font> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E3:51_6_1"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="lattice8.html#E4:51_6_1"><i><font color="Green" title="E52">A29</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">d1</font>
<b>by </b><i><a class="txt" href="lattice8.html#E18:51"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="lattice8.html#E9:51_6_1"><i><font color="Green" title="E54">A31</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E10:51_6_1"><i><font color="Green" title="E55">A32</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E12:51_6_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:51_6"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:51_6"/><i><font color="Green" title="E51">A33</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>
<b>consider </b><font color="Maroon" title="c16">A1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c17">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c16">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c18">Aq1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c19">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c18">Aq1</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:51_6"/>
<font color="Maroon" title="c18">Aq1</font>,<font color="Maroon" title="c19">dq1</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c16">A1</font>,<font color="Maroon" title="c17">d1</font>
 <b>and </b><br/><a NAME="E7:51_6"/><i><font color="Green" title="E52">A34</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">A1</font>,<font color="Maroon" title="c17">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">Aq1</font>,<font color="Maroon" title="c19">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E8:51_6"/><i><font color="Green" title="E53">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">d1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:51_6"><i><font color="Green" title="E52">A34</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:51_6"/><i><font color="Green" title="E54">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">A1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:51_6"><i><font color="Green" title="E52">A34</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:51_6"/>
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E11:51_6"/><b>then </b><i><font color="Green" title="E55">A37</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c11">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E17:51"><i><font color="Green" title="E47">A21</font></i></a>, <a class="ref" href="funct_6.html#T89" title="FUNCT_6:th.89">FUNCT_6:89</a></i>;<br/>
<font color="Maroon" title="c14">x</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">d1</font>
<b>by </b><i><a class="txt" href="lattice8.html#E4:51_6"><i><font color="Green" title="E51">A33</font></i></a>, <a class="txt" href="lattice8.html#E9:51_6"><i><font color="Green" title="E54">A36</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E18:51"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="lattice8.html#E8:51_6"><i><font color="Green" title="E53">A35</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="lattice8.html#E17:51"><i><font color="Green" title="E47">A21</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>
;<br/>
<b>hence </b><a NAME="E13:51_6"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c11">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E11:51_6"><i><font color="Green" title="E55">A37</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:51"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:51_7"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_7_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">J</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">X</font><b> such that </b><br/><a NAME="E3:51_7_1"/><i><font color="Green" title="E51">A38</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c15">J</font>,<font color="Maroon" title="c15">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E4:51_7_1"/><i><font color="Green" title="E52">A39</font></i>: 
<font color="Maroon" title="c15">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c16">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:51_7_1"/><i><font color="Green" title="E53">A40</font></i>: 
<font color="Maroon" title="c15">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E4:51_7_1"><i><font color="Green" title="E52">A39</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:51_7_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="lattice8.html#E3:51_7_1"><i><font color="Green" title="E51">A38</font></i></a>, <a class="txt" href="lattice8.html#E6:51_7_1"><i><font color="Green" title="E53">A40</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:51_7"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:51_7"/><i><font color="Green" title="E51">A41</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>
<a NAME="E5:51_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
;<br/>
<b>hence </b><a NAME="E6:51_7"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="lattice8.html#E4:51_7"><i><font color="Green" title="E51">A41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">FS</font>,<font color="Maroon" title="c7">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c11">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:51"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="lattice8.html#E13:51"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="lattice8.html#E22:51"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="lattice5.html#T3" title="LATTICE5:th.3">LATTICE5:3</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">FD</font>
<b>by </b><i><a class="txt" href="lattice8.html#E19:51"><i><font color="Green" title="E49">A23</font></i></a>, <a class="ref" href="lattice5.html#T1" title="LATTICE5:th.1">LATTICE5:1</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">FD</font> = <font color="Maroon" title="c9">FD</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c7">FS</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="lattice8.html#E9:51"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="lattice8.html#E11:51"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E26:51"/><i><font color="Green" title="E51">A42</font></i>: 
<font color="Maroon" title="c14">FD</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">FS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a><br/>

<b>consider </b><font color="Maroon" title="c16">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:51_9"/><i><font color="Green" title="E52">A43</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:51"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:51_9"/><i><font color="Green" title="E53">A44</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E2:51_9"><i><font color="Green" title="E52">A43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">A1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c19">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c18">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c20">Aq1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c21">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c20">Aq1</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:51_9"/>
<font color="Maroon" title="c20">Aq1</font>,<font color="Maroon" title="c21">dq1</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c18">A1</font>,<font color="Maroon" title="c19">d1</font>
 <b>and </b><br/><a NAME="E7:51_9"/><i><font color="Green" title="E54">A45</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">A1</font>,<font color="Maroon" title="c19">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">Aq1</font>,<font color="Maroon" title="c21">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E8:51_9"/><i><font color="Green" title="E55">A46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:51_9"><i><font color="Green" title="E54">A45</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">x'</font> = <font color="Maroon" title="c15">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">A1</font> <b>by </b><i><a class="txt" href="lattice8.html#E2:51_9"><i><font color="Green" title="E52">A43</font></i></a>, <a class="txt" href="lattice8.html#E4:51_9"><i><font color="Green" title="E53">A44</font></i></a>, <a class="txt" href="lattice8.html#E7:51_9"><i><font color="Green" title="E54">A45</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:51_9"/>
<font color="Maroon" title="c19">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E8:51_9"><i><font color="Green" title="E55">A46</font></i></a></i>;<br/>
<a NAME="E11:51_9"/><b>then </b><i><font color="Green" title="E56">A47</font></i>: 
<font color="Maroon" title="c19">d1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:51_9"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">A1</font>,<font color="Maroon" title="c18">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c15">x</font> = 
<font color="Maroon" title="c19">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">x'</font>,<font color="Maroon" title="c22">x'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E11:51_9"><i><font color="Green" title="E56">A47</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c19">d1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c22">x'</font>,<font color="Maroon" title="c22">x'</font>

<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E27:51"/><i><font color="Green" title="E52">A48</font></i>: 
<font color="Maroon" title="c14">FD</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">FS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">LATTICE5:def 6</a><br/>

<b>consider </b><font color="Maroon" title="c17">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:51_10"/><i><font color="Green" title="E53">A49</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">x1</font> &amp; <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:51"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:51_10"/><i><font color="Green" title="E54">A50</font></i>: 
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E2:51_10"><i><font color="Green" title="E53">A49</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">A1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c20">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c21">Aq1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c22">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c21">Aq1</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:51_10"/>
<font color="Maroon" title="c21">Aq1</font>,<font color="Maroon" title="c22">dq1</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c20">d1</font>
 <b>and </b><br/><a NAME="E7:51_10"/><i><font color="Green" title="E55">A51</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c20">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">Aq1</font>,<font color="Maroon" title="c22">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E8:51_10"/><i><font color="Green" title="E56">A52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">d1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:51_10"><i><font color="Green" title="E55">A51</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:51_10"/><i><font color="Green" title="E57">A53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">A1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:51_10"><i><font color="Green" title="E55">A51</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:51_10"/>
<font color="Maroon" title="c20">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E8:51_10"><i><font color="Green" title="E56">A52</font></i></a></i>;<br/>
<a NAME="E11:51_10"/><b>then </b><i><font color="Green" title="E58">A54</font></i>: 
<font color="Maroon" title="c20">d1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:51_10"/><i><font color="Green" title="E59">A55</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">A1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:51_10"><i><font color="Green" title="E53">A49</font></i></a>, <a class="txt" href="lattice8.html#E4:51_10"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="lattice8.html#E7:51_10"><i><font color="Green" title="E55">A51</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:51_10"/><i><font color="Green" title="E60">A56</font></i>: 
( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y1</font> &amp; <font color="Maroon" title="c23">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:51"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E16:51_10"/><i><font color="Green" title="E61">A57</font></i>: 
<font color="Maroon" title="c23">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E14:51_10"><i><font color="Green" title="E60">A56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">A2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c26">d2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c25">A2</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c27">Aq2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c28">dq2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c27">Aq2</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E18:51_10"/>
<font color="Maroon" title="c27">Aq2</font>,<font color="Maroon" title="c28">dq2</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c25">A2</font>,<font color="Maroon" title="c26">d2</font>
 <b>and </b><br/><a NAME="E19:51_10"/><i><font color="Green" title="E62">A58</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">A2</font>,<font color="Maroon" title="c26">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">Aq2</font>,<font color="Maroon" title="c28">dq2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E20:51_10"/><i><font color="Green" title="E63">A59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">d2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E19:51_10"><i><font color="Green" title="E62">A58</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E21:51_10"/><i><font color="Green" title="E64">A60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">A2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E19:51_10"><i><font color="Green" title="E62">A58</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E22:51_10"/>
<font color="Maroon" title="c26">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E20:51_10"><i><font color="Green" title="E63">A59</font></i></a></i>;<br/>
<a NAME="E23:51_10"/><b>then </b><i><font color="Green" title="E65">A61</font></i>: 
<font color="Maroon" title="c26">d2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E24:51_10"/><i><font color="Green" title="E66">A62</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E14:51_10"><i><font color="Green" title="E60">A56</font></i></a>, <a class="txt" href="lattice8.html#E16:51_10"><i><font color="Green" title="E61">A57</font></i></a>, <a class="txt" href="lattice8.html#E19:51_10"><i><font color="Green" title="E62">A58</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_10_1"/>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">l</font> or <font color="Maroon" title="c24">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_10_1_1"><b>suppose </b></a><a NAME="E1:51_10_1_1"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">l</font>
 ;<br/><div class="add"><a NAME="E2:51_10_1_1"/><b>then </b>
<font color="Maroon" title="c19">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">A2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:51_10"><i><font color="Green" title="E57">A53</font></i></a>, <a class="txt" href="lattice8.html#E21:51_10"><i><font color="Green" title="E64">A60</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self" title="LATTICE8:th.31">Th31</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">x'</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c30">y'</font> = <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c25">A2</font> <b>by </b><i><a class="txt" href="lattice8.html#E12:51_10"><i><font color="Green" title="E59">A55</font></i></a>, <a class="txt" href="lattice8.html#E14:51_10"><i><font color="Green" title="E60">A56</font></i></a>, <a class="txt" href="lattice8.html#E16:51_10"><i><font color="Green" title="E61">A57</font></i></a>, <a class="txt" href="lattice8.html#E19:51_10"><i><font color="Green" title="E62">A58</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/><a NAME="E4:51_10_1_1"/><i><font color="Green" title="E67">A63</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c25">A2</font>,<font color="Maroon" title="c25">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> = 
<font color="Maroon" title="c26">d2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">x'</font>,<font color="Maroon" title="c30">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E23:51_10"><i><font color="Green" title="E65">A61</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c26">d2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c29">x'</font>,<font color="Maroon" title="c30">y'</font>

<br/>.= 
<font color="Maroon" title="c26">d2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c30">y'</font>,<font color="Maroon" title="c29">x'</font>
<b>by </b><i><a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">LATTICE5:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c14">FD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">y'</font>,<font color="Maroon" title="c29">x'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E23:51_10"><i><font color="Green" title="E65">A61</font></i></a>, <a class="txt" href="lattice8.html#E4:51_10_1_1"><i><font color="Green" title="E67">A63</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c15">x</font>

;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_10_1_2"><b>suppose </b></a><a NAME="E1:51_10_1_2"/>
<font color="Maroon" title="c24">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font>
 ;<br/><div class="add"><a NAME="E2:51_10_1_2"/><b>then </b>
<font color="Maroon" title="c25">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">A1</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:51_10"><i><font color="Green" title="E57">A53</font></i></a>, <a class="txt" href="lattice8.html#E21:51_10"><i><font color="Green" title="E64">A60</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self" title="LATTICE8:th.31">Th31</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">x'</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c30">y'</font> = <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">A1</font> <b>by </b><i><a class="txt" href="lattice8.html#E2:51_10"><i><font color="Green" title="E53">A49</font></i></a>, <a class="txt" href="lattice8.html#E4:51_10"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="lattice8.html#E7:51_10"><i><font color="Green" title="E55">A51</font></i></a>, <a class="txt" href="lattice8.html#E24:51_10"><i><font color="Green" title="E66">A62</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/><a NAME="E4:51_10_1_2"/><i><font color="Green" title="E67">A64</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c19">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> = 
<font color="Maroon" title="c20">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">x'</font>,<font color="Maroon" title="c30">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E11:51_10"><i><font color="Green" title="E58">A54</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c20">d1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c29">x'</font>,<font color="Maroon" title="c30">y'</font>

<br/>.= 
<font color="Maroon" title="c20">d1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c30">y'</font>,<font color="Maroon" title="c29">x'</font>
<b>by </b><i><a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">LATTICE5:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c14">FD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">y'</font>,<font color="Maroon" title="c29">x'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E11:51_10"><i><font color="Green" title="E58">A54</font></i></a>, <a class="txt" href="lattice8.html#E4:51_10_1_2"><i><font color="Green" title="E67">A64</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c15">x</font>

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

</div><b>end;</b></div>
<a NAME="E28:51"/>
<font color="Maroon" title="c14">FD</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font>, <font color="Maroon" title="c17">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">FS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a><br/>

<b>consider </b><font color="Maroon" title="c18">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:51_11"/><i><font color="Green" title="E53">A65</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> &amp; <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:51"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:51_11"/><i><font color="Green" title="E54">A66</font></i>: 
<font color="Maroon" title="c18">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E2:51_11"><i><font color="Green" title="E53">A65</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">A1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c21">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c20">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c22">Aq1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c23">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c22">Aq1</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:51_11"/>
<font color="Maroon" title="c22">Aq1</font>,<font color="Maroon" title="c23">dq1</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c20">A1</font>,<font color="Maroon" title="c21">d1</font>
 <b>and </b><br/><a NAME="E7:51_11"/><i><font color="Green" title="E55">A67</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">A1</font>,<font color="Maroon" title="c21">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">Aq1</font>,<font color="Maroon" title="c23">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E8:51_11"/><i><font color="Green" title="E56">A68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">d1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:51_11"><i><font color="Green" title="E55">A67</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:51_11"/><i><font color="Green" title="E57">A69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">A1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:51_11"><i><font color="Green" title="E55">A67</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E10:51_11"/>
<font color="Maroon" title="c21">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E8:51_11"><i><font color="Green" title="E56">A68</font></i></a></i>;<br/>
<a NAME="E11:51_11"/><b>then </b><i><font color="Green" title="E58">A70</font></i>: 
<font color="Maroon" title="c21">d1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:51_11"/><i><font color="Green" title="E59">A71</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">A1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:51_11"><i><font color="Green" title="E53">A65</font></i></a>, <a class="txt" href="lattice8.html#E4:51_11"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="lattice8.html#E7:51_11"><i><font color="Green" title="E55">A67</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E13:51_11"/><i><font color="Green" title="E60">A72</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c20">A1</font>,<font color="Maroon" title="c20">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:51_11"/><i><font color="Green" title="E61">A73</font></i>: 
( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">y1</font> &amp; <font color="Maroon" title="c24">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:51"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E17:51_11"/><i><font color="Green" title="E62">A74</font></i>: 
<font color="Maroon" title="c24">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E15:51_11"><i><font color="Green" title="E61">A73</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">A2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c27">d2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c26">A2</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c28">Aq2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c29">dq2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c28">Aq2</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E19:51_11"/>
<font color="Maroon" title="c28">Aq2</font>,<font color="Maroon" title="c29">dq2</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c26">A2</font>,<font color="Maroon" title="c27">d2</font>
 <b>and </b><br/><a NAME="E20:51_11"/><i><font color="Green" title="E63">A75</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">A2</font>,<font color="Maroon" title="c27">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">Aq2</font>,<font color="Maroon" title="c29">dq2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E21:51_11"/><i><font color="Green" title="E64">A76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">l</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">d2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E20:51_11"><i><font color="Green" title="E63">A75</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E22:51_11"/><i><font color="Green" title="E65">A77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">A2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E20:51_11"><i><font color="Green" title="E63">A75</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E23:51_11"/>
<font color="Maroon" title="c27">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E21:51_11"><i><font color="Green" title="E64">A76</font></i></a></i>;<br/>
<a NAME="E24:51_11"/><b>then </b><i><font color="Green" title="E66">A78</font></i>: 
<font color="Maroon" title="c27">d2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E25:51_11"/><i><font color="Green" title="E67">A79</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">A2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E15:51_11"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="lattice8.html#E17:51_11"><i><font color="Green" title="E62">A74</font></i></a>, <a class="txt" href="lattice8.html#E20:51_11"><i><font color="Green" title="E63">A75</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E26:51_11"/><i><font color="Green" title="E68">A80</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c26">A2</font>,<font color="Maroon" title="c26">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c30">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E28:51_11"/><i><font color="Green" title="E69">A81</font></i>: 
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">z1</font> &amp; <font color="Maroon" title="c30">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:51"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c31">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E30:51_11"/><i><font color="Green" title="E70">A82</font></i>: 
<font color="Maroon" title="c30">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice8.html#E28:51_11"><i><font color="Green" title="E69">A81</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c32">A3</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c33">d3</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c32">A3</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c34">Aq3</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c35">dq3</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c34">Aq3</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E32:51_11"/>
<font color="Maroon" title="c34">Aq3</font>,<font color="Maroon" title="c35">dq3</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c32">A3</font>,<font color="Maroon" title="c33">d3</font>
 <b>and </b><br/><a NAME="E33:51_11"/><i><font color="Green" title="E71">A83</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">A3</font>,<font color="Maroon" title="c33">d3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">Aq3</font>,<font color="Maroon" title="c35">dq3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self" title="LATTICE8:def.12">Def12</a></i>;<br/>
<a NAME="E34:51_11"/><i><font color="Green" title="E72">A84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">d3</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E33:51_11"><i><font color="Green" title="E71">A83</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E35:51_11"/><i><font color="Green" title="E73">A85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">A3</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E33:51_11"><i><font color="Green" title="E71">A83</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E36:51_11"/>
<font color="Maroon" title="c33">d3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E34:51_11"><i><font color="Green" title="E72">A84</font></i></a></i>;<br/>
<a NAME="E37:51_11"/><b>then </b><i><font color="Green" title="E74">A86</font></i>: 
<font color="Maroon" title="c33">d3</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c14">FD</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E38:51_11"/><i><font color="Green" title="E75">A87</font></i>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">A3</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E28:51_11"><i><font color="Green" title="E69">A81</font></i></a>, <a class="txt" href="lattice8.html#E30:51_11"><i><font color="Green" title="E70">A82</font></i></a>, <a class="txt" href="lattice8.html#E33:51_11"><i><font color="Green" title="E71">A83</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E39:51_11"/><i><font color="Green" title="E76">A88</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">d3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c32">A3</font>,<font color="Maroon" title="c32">A3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_11_1"/>
( <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">l</font> or <font color="Maroon" title="c25">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_11_1_1"><b>suppose </b></a><a NAME="E1:51_11_1_1"/>
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">l</font>
 ;<br/><div class="add"><a NAME="E2:51_11_1_1"/><b>then </b><i><font color="Green" title="E77">A89</font></i>: 
<font color="Maroon" title="c20">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">A2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:51_11"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="lattice8.html#E22:51_11"><i><font color="Green" title="E65">A77</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self" title="LATTICE8:th.31">Th31</a></i>;<br/><b>thus </b><a NAME="E3:51_11_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_11_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_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="E1:51_11_1_1_1_1"/>
( <font color="Maroon" title="c25">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font> or <font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">l</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_11_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:51_11_1_1_1_1_1"/>
<font color="Maroon" title="c25">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font>
 ;<br/><div class="add"><a NAME="E2:51_11_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E78">A90</font></i>: 
<font color="Maroon" title="c26">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c32">A3</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E22:51_11"><i><font color="Green" title="E65">A77</font></i></a>, <a class="txt" href="lattice8.html#E35:51_11"><i><font color="Green" title="E73">A85</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self" title="LATTICE8:th.31">Th31</a></i>;<br/><a NAME="E3:51_11_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c32">A3</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:51_11_1_1"><i><font color="Green" title="E77">A89</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c36">x'</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c37">y'</font> = <font color="Maroon" title="c16">y</font>, <font color="Maroon" title="c38">z'</font> = <font color="Maroon" title="c17">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c32">A3</font> <b>by </b><i><a class="txt" href="lattice8.html#E12:51_11"><i><font color="Green" title="E59">A71</font></i></a>, <a class="txt" href="lattice8.html#E25:51_11"><i><font color="Green" title="E67">A79</font></i></a>, <a class="txt" href="lattice8.html#E28:51_11"><i><font color="Green" title="E69">A81</font></i></a>, <a class="txt" href="lattice8.html#E30:51_11"><i><font color="Green" title="E70">A82</font></i></a>, <a class="txt" href="lattice8.html#E33:51_11"><i><font color="Green" title="E71">A83</font></i></a>, <a class="txt" href="lattice8.html#E2:51_11_1_1_1_1_1"><i><font color="Green" title="E78">A90</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/><i><font color="Green" title="E79">A91</font></i>: <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> = 
<font color="Maroon" title="c33">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c37">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E37:51_11"><i><font color="Green" title="E74">A86</font></i></a>, <a class="txt" href="lattice8.html#E39:51_11"><i><font color="Green" title="E76">A88</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c33">d3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c37">y'</font>

;<br/><i><font color="Green" title="E80">A92</font></i>: <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font> = 
<font color="Maroon" title="c33">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c37">y'</font>,<font color="Maroon" title="c38">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E37:51_11"><i><font color="Green" title="E74">A86</font></i></a>, <a class="txt" href="lattice8.html#E39:51_11"><i><font color="Green" title="E76">A88</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c33">d3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c37">y'</font>,<font color="Maroon" title="c38">z'</font>

;<br/><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font> = 
<font color="Maroon" title="c33">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c38">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E37:51_11"><i><font color="Green" title="E74">A86</font></i></a>, <a class="txt" href="lattice8.html#E39:51_11"><i><font color="Green" title="E76">A88</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c33">d3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c38">z'</font>

;<br/><b>hence </b><a NAME="E8:51_11_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:51_11_1_1_1_1_1"><i><font color="Green" title="E79">A91</font></i></a>, <a class="txt" href="lattice8.html#E6:51_11_1_1_1_1_1"><i><font color="Green" title="E80">A92</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_11_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:51_11_1_1_1_1_2"/>
<font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">l</font>
 ;<br/><div class="add"><a NAME="E2:51_11_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c32">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">A2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E22:51_11"><i><font color="Green" title="E65">A77</font></i></a>, <a class="txt" href="lattice8.html#E35:51_11"><i><font color="Green" title="E73">A85</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self" title="LATTICE8:th.31">Th31</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c36">x'</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c37">y'</font> = <font color="Maroon" title="c16">y</font>, <font color="Maroon" title="c38">z'</font> = <font color="Maroon" title="c17">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c26">A2</font> <b>by </b><i><a class="txt" href="lattice8.html#E12:51_11"><i><font color="Green" title="E59">A71</font></i></a>, <a class="txt" href="lattice8.html#E15:51_11"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="lattice8.html#E17:51_11"><i><font color="Green" title="E62">A74</font></i></a>, <a class="txt" href="lattice8.html#E20:51_11"><i><font color="Green" title="E63">A75</font></i></a>, <a class="txt" href="lattice8.html#E38:51_11"><i><font color="Green" title="E75">A87</font></i></a>, <a class="txt" href="lattice8.html#E2:51_11_1_1"><i><font color="Green" title="E77">A89</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/><i><font color="Green" title="E78">A93</font></i>: <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> = 
<font color="Maroon" title="c27">d2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c37">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E24:51_11"><i><font color="Green" title="E66">A78</font></i></a>, <a class="txt" href="lattice8.html#E26:51_11"><i><font color="Green" title="E68">A80</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c27">d2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c37">y'</font>

;<br/><i><font color="Green" title="E79">A94</font></i>: <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font> = 
<font color="Maroon" title="c27">d2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c37">y'</font>,<font color="Maroon" title="c38">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E24:51_11"><i><font color="Green" title="E66">A78</font></i></a>, <a class="txt" href="lattice8.html#E26:51_11"><i><font color="Green" title="E68">A80</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c27">d2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c37">y'</font>,<font color="Maroon" title="c38">z'</font>

;<br/><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font> = 
<font color="Maroon" title="c27">d2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c38">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E24:51_11"><i><font color="Green" title="E66">A78</font></i></a>, <a class="txt" href="lattice8.html#E26:51_11"><i><font color="Green" title="E68">A80</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c27">d2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c38">z'</font>

;<br/><b>hence </b><a NAME="E7:51_11_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:51_11_1_1_1_1_2"><i><font color="Green" title="E78">A93</font></i></a>, <a class="txt" href="lattice8.html#E5:51_11_1_1_1_1_2"><i><font color="Green" title="E79">A94</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_11_1_2"><b>suppose </b></a><a NAME="E1:51_11_1_2"/>
<font color="Maroon" title="c25">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 ;<br/><div class="add"><a NAME="E2:51_11_1_2"/><b>then </b><i><font color="Green" title="E77">A95</font></i>: 
<font color="Maroon" title="c26">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">A1</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:51_11"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="lattice8.html#E22:51_11"><i><font color="Green" title="E65">A77</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self" title="LATTICE8:th.31">Th31</a></i>;<br/><b>thus </b><a NAME="E3:51_11_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_11_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_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="E1:51_11_1_2_1_1"/>
( <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font> or <font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_11_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:51_11_1_2_1_1_1"/>
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font>
 ;<br/><div class="add"><a NAME="E2:51_11_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E78">A96</font></i>: 
<font color="Maroon" title="c20">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c32">A3</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:51_11"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="lattice8.html#E35:51_11"><i><font color="Green" title="E73">A85</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self" title="LATTICE8:th.31">Th31</a></i>;<br/><a NAME="E3:51_11_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c26">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c32">A3</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:51_11_1_2"><i><font color="Green" title="E77">A95</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c36">x'</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c37">y'</font> = <font color="Maroon" title="c16">y</font>, <font color="Maroon" title="c38">z'</font> = <font color="Maroon" title="c17">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c32">A3</font> <b>by </b><i><a class="txt" href="lattice8.html#E12:51_11"><i><font color="Green" title="E59">A71</font></i></a>, <a class="txt" href="lattice8.html#E25:51_11"><i><font color="Green" title="E67">A79</font></i></a>, <a class="txt" href="lattice8.html#E28:51_11"><i><font color="Green" title="E69">A81</font></i></a>, <a class="txt" href="lattice8.html#E30:51_11"><i><font color="Green" title="E70">A82</font></i></a>, <a class="txt" href="lattice8.html#E33:51_11"><i><font color="Green" title="E71">A83</font></i></a>, <a class="txt" href="lattice8.html#E2:51_11_1_2_1_1_1"><i><font color="Green" title="E78">A96</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/><i><font color="Green" title="E79">A97</font></i>: <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> = 
<font color="Maroon" title="c33">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c37">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E37:51_11"><i><font color="Green" title="E74">A86</font></i></a>, <a class="txt" href="lattice8.html#E39:51_11"><i><font color="Green" title="E76">A88</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c33">d3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c37">y'</font>

;<br/><i><font color="Green" title="E80">A98</font></i>: <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font> = 
<font color="Maroon" title="c33">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c37">y'</font>,<font color="Maroon" title="c38">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E37:51_11"><i><font color="Green" title="E74">A86</font></i></a>, <a class="txt" href="lattice8.html#E39:51_11"><i><font color="Green" title="E76">A88</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c33">d3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c37">y'</font>,<font color="Maroon" title="c38">z'</font>

;<br/><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font> = 
<font color="Maroon" title="c33">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c38">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E37:51_11"><i><font color="Green" title="E74">A86</font></i></a>, <a class="txt" href="lattice8.html#E39:51_11"><i><font color="Green" title="E76">A88</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c33">d3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c38">z'</font>

;<br/><b>hence </b><a NAME="E8:51_11_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:51_11_1_2_1_1_1"><i><font color="Green" title="E79">A97</font></i></a>, <a class="txt" href="lattice8.html#E6:51_11_1_2_1_1_1"><i><font color="Green" title="E80">A98</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_11_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:51_11_1_2_1_1_2"/>
<font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 ;<br/><div class="add"><a NAME="E2:51_11_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c32">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">A1</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:51_11"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="lattice8.html#E35:51_11"><i><font color="Green" title="E73">A85</font></i></a>, <a class="ref" href="lattice8.html#T31" target="_self" title="LATTICE8:th.31">Th31</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c36">x'</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c37">y'</font> = <font color="Maroon" title="c16">y</font>, <font color="Maroon" title="c38">z'</font> = <font color="Maroon" title="c17">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c20">A1</font> <b>by </b><i><a class="txt" href="lattice8.html#E2:51_11"><i><font color="Green" title="E53">A65</font></i></a>, <a class="txt" href="lattice8.html#E4:51_11"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="lattice8.html#E7:51_11"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="lattice8.html#E25:51_11"><i><font color="Green" title="E67">A79</font></i></a>, <a class="txt" href="lattice8.html#E38:51_11"><i><font color="Green" title="E75">A87</font></i></a>, <a class="txt" href="lattice8.html#E2:51_11_1_2"><i><font color="Green" title="E77">A95</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/><i><font color="Green" title="E78">A99</font></i>: <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> = 
<font color="Maroon" title="c21">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c37">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E11:51_11"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="lattice8.html#E13:51_11"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c21">d1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c37">y'</font>

;<br/><i><font color="Green" title="E79">A100</font></i>: <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font> = 
<font color="Maroon" title="c21">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c37">y'</font>,<font color="Maroon" title="c38">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E11:51_11"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="lattice8.html#E13:51_11"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c21">d1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c37">y'</font>,<font color="Maroon" title="c38">z'</font>

;<br/><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font> = 
<font color="Maroon" title="c21">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c38">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E11:51_11"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="lattice8.html#E13:51_11"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c21">d1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c36">x'</font>,<font color="Maroon" title="c38">z'</font>

;<br/><b>hence </b><a NAME="E7:51_11_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c17">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c14">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:51_11_1_2_1_1_2"><i><font color="Green" title="E78">A99</font></i></a>, <a class="txt" href="lattice8.html#E5:51_11_1_2_1_1_2"><i><font color="Green" title="E79">A100</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<b>hence </b><a NAME="E29:51"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  is   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c3">FS</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E26:51"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="lattice8.html#E27:51"><i><font color="Green" title="E52">A48</font></i></a></i>;<br/>


</div>
