<?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="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="lattice5.html#M2" title="LATTICE5:mode.2">ExtensionSeq</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="E1:81"/><i><font color="Green" title="E61">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="E2:81"/>
<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> 0</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="E5:81"/><i><font color="Green" title="E62">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="81_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" 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="E1:81_1_1"/><i><font color="Green" title="E63">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="E2:81_1_1"/><i><font color="Green" title="E64">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="E4:81_1_1"/><i><font color="Green" title="E65">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="lattice5.html#E1:81_1_1"><i><font color="Green" title="E63">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="E6:81_1_1"/><i><font color="Green" title="E66">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="lattice5.html#E2:81_1_1"><i><font color="Green" title="E64">A4</font></i></a></i>;<br/><a NAME="E7:81_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="E8:81_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="lattice5.html#E4:81_1_1"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="lattice5.html#E6:81_1_1"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="lattice5.html#T42" target="_self" title="LATTICE5:th.42">Th42</a></i>;<br/><b>hence </b><a NAME="E9:81_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="E2:81_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="E6:81"/>
<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="81_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="E1:81_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="E3:81_2"/><i><font color="Green" title="E63">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="E5:81_2"/>
<font color="Maroon" title="c12">Aq</font>,<font color="Maroon" title="c13">dq</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">d'</font>
 <b>and </b><br/><a NAME="E6:81_2"/><i><font color="Green" title="E64">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E7:81_2"/><i><font color="Green" title="E65">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="lattice5.html#E3:81_2"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="lattice5.html#E6:81_2"><i><font color="Green" title="E64">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="E8:81_2"/><i><font color="Green" title="E66">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="E9:81_2"/><i><font color="Green" title="E67">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="E10:81_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="lattice5.html#E6:81_2"><i><font color="Green" title="E64">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="E11:81_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="E12:81_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="lattice5.html#E1:81"><i><font color="Green" title="E61">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="E13:81_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="lattice5.html#E9:81_2"><i><font color="Green" title="E67">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="E14:81_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="lattice5.html#E7:81_2"><i><font color="Green" title="E65">A9</font></i></a>, <a class="txt" href="lattice5.html#E8:81_2"><i><font color="Green" title="E66">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="E7:81"/><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="lattice5.html#E5:81"><i><font color="Green" title="E62">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="E9:81"/><i><font color="Green" title="E63">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="E10:81"/>
 <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="E11:81"/><i><font color="Green" title="E64">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="lattice5.html#E9:81"><i><font color="Green" title="E63">A12</font></i></a></i>;<br/>
<a NAME="E13:81"/><i><font color="Green" title="E65">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="81_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" 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="E1:81_3_1"/><i><font color="Green" title="E66">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="E2:81_3_1"/><i><font color="Green" title="E67">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="E4:81_3_1"/><i><font color="Green" title="E68">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="lattice5.html#E1:81_3_1"><i><font color="Green" title="E66">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="E6:81_3_1"/><i><font color="Green" title="E69">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="lattice5.html#E2:81_3_1"><i><font color="Green" title="E67">A16</font></i></a></i>;<br/><a NAME="E7:81_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="E8:81_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="lattice5.html#E4:81_3_1"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="lattice5.html#E6:81_3_1"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</a></i>;<br/><b>hence </b><a NAME="E9:81_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="E2:81_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="E14:81"/><i><font color="Green" title="E66">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="E15:81"/><i><font color="Green" title="E67">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="E17:81"/><i><font color="Green" title="E68">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="E18:81"/><i><font color="Green" title="E69">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="lattice5.html#E14:81"><i><font color="Green" title="E66">A19</font></i></a>, <a class="txt" href="lattice5.html#E15:81"><i><font color="Green" title="E67">A20</font></i></a>);<br/></i>
<a NAME="E19:81"/><i><font color="Green" title="E70">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="81_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_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="81_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="E1:81_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="E3:81_4_1"/><i><font color="Green" title="E71">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="lattice5.html#E17:81"><i><font color="Green" title="E68">A21</font></i></a>, <a class="txt" href="lattice5.html#E3:81_4_1"><i><font color="Green" title="E71">A24</font></i></a></i>;<br/>
<a NAME="E5:81_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="lattice5.html#E18:81"><i><font color="Green" title="E69">A22</font></i></a>, <a class="txt" href="lattice5.html#E3:81_4_1"><i><font color="Green" title="E71">A24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:81_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="E2:81_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="E4:81_4"/><i><font color="Green" title="E71">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="E5:81_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="lattice5.html#E18:81"><i><font color="Green" title="E69">A22</font></i></a>, <a class="txt" href="lattice5.html#E4:81_4"><i><font color="Green" title="E71">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:81_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="lattice5.html#E17:81"><i><font color="Green" title="E68">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="E20:81"/>
<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="81_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="E1:81_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="lattice5.html#E17:81"><i><font color="Green" title="E68">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="E4:81_5"/>
<font color="Maroon" title="c15">Aq1</font>,<font color="Maroon" title="c16">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c13">A1</font>,<font color="Maroon" title="c14">d1</font>
 <b>and </b><br/><a NAME="E5:81_5"/><i><font color="Green" title="E71">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E6:81_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="lattice5.html#E5:81_5"><i><font color="Green" title="E71">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="E7:81_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="lattice5.html#E18:81"><i><font color="Green" title="E69">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="E22:81"/><i><font color="Green" title="E71">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="81_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_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="81_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="E1:81_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="E3:81_6_1"/><i><font color="Green" title="E72">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="E4:81_6_1"/><i><font color="Green" title="E73">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="lattice5.html#E3:81_6_1"><i><font color="Green" title="E72">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="lattice5.html#E17:81"><i><font color="Green" title="E68">A21</font></i></a>, <a class="txt" href="lattice5.html#E3:81_6_1"><i><font color="Green" title="E72">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="E7:81_6_1"/>
<font color="Maroon" title="c19">Aq1</font>,<font color="Maroon" title="c20">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c18">d1</font>
 <b>and </b><br/><a NAME="E8:81_6_1"/><i><font color="Green" title="E74">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E9:81_6_1"/><i><font color="Green" title="E75">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="lattice5.html#E8:81_6_1"><i><font color="Green" title="E74">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="E10:81_6_1"/><i><font color="Green" title="E76">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="lattice5.html#E8:81_6_1"><i><font color="Green" title="E74">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="lattice5.html#E3:81_6_1"><i><font color="Green" title="E72">A28</font></i></a>, <a class="txt" href="lattice5.html#E4:81_6_1"><i><font color="Green" title="E73">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="lattice5.html#E18:81"><i><font color="Green" title="E69">A22</font></i></a>, <a class="txt" href="lattice5.html#E9:81_6_1"><i><font color="Green" title="E75">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="lattice5.html#E10:81_6_1"><i><font color="Green" title="E76">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="E12:81_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="E2:81_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="E4:81_6"/><i><font color="Green" title="E72">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="E6:81_6"/>
<font color="Maroon" title="c18">Aq1</font>,<font color="Maroon" title="c19">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c16">A1</font>,<font color="Maroon" title="c17">d1</font>
 <b>and </b><br/><a NAME="E7:81_6"/><i><font color="Green" title="E73">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E8:81_6"/><i><font color="Green" title="E74">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="lattice5.html#E7:81_6"><i><font color="Green" title="E73">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="E9:81_6"/><i><font color="Green" title="E75">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="lattice5.html#E7:81_6"><i><font color="Green" title="E73">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="E10:81_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="E11:81_6"/><b>then </b><i><font color="Green" title="E76">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="lattice5.html#E17:81"><i><font color="Green" title="E68">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="lattice5.html#E4:81_6"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="lattice5.html#E9:81_6"><i><font color="Green" title="E75">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="lattice5.html#E18:81"><i><font color="Green" title="E69">A22</font></i></a>, <a class="txt" href="lattice5.html#E8:81_6"><i><font color="Green" title="E74">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="lattice5.html#E17:81"><i><font color="Green" title="E68">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="E13:81_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="lattice5.html#E11:81_6"><i><font color="Green" title="E76">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="E23:81"/>
<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="81_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_7"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" 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="81_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="E1:81_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="E3:81_7_1"/><i><font color="Green" title="E72">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="E4:81_7_1"/><i><font color="Green" title="E73">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="E6:81_7_1"/><i><font color="Green" title="E74">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="lattice5.html#E4:81_7_1"><i><font color="Green" title="E73">A39</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:81_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="lattice5.html#E3:81_7_1"><i><font color="Green" title="E72">A38</font></i></a>, <a class="txt" href="lattice5.html#E6:81_7_1"><i><font color="Green" title="E74">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="E2:81_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="E4:81_7"/><i><font color="Green" title="E72">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="E5:81_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="E6:81_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="lattice5.html#E4:81_7"><i><font color="Green" title="E72">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="lattice5.html#E1:81"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="lattice5.html#E13:81"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="lattice5.html#E22:81"><i><font color="Green" title="E71">A27</font></i></a>, <a class="ref" href="lattice5.html#T3" target="_self" title="LATTICE5:th.3">Th3</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="lattice5.html#E19:81"><i><font color="Green" title="E70">A23</font></i></a>, <a class="ref" href="lattice5.html#T1" target="_self" title="LATTICE5:th.1">Th1</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="lattice5.html#E9:81"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="lattice5.html#E11:81"><i><font color="Green" title="E64">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="E26:81"/><i><font color="Green" title="E72">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="81_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" target="_self" 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="E2:81_9"/><i><font color="Green" title="E73">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="lattice5.html#E1:81"><i><font color="Green" title="E61">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="E4:81_9"/><i><font color="Green" title="E74">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="lattice5.html#E2:81_9"><i><font color="Green" title="E73">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="E6:81_9"/>
<font color="Maroon" title="c20">Aq1</font>,<font color="Maroon" title="c21">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c18">A1</font>,<font color="Maroon" title="c19">d1</font>
 <b>and </b><br/><a NAME="E7:81_9"/><i><font color="Green" title="E75">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E8:81_9"/><i><font color="Green" title="E76">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="lattice5.html#E7:81_9"><i><font color="Green" title="E75">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="lattice5.html#E2:81_9"><i><font color="Green" title="E73">A43</font></i></a>, <a class="txt" href="lattice5.html#E4:81_9"><i><font color="Green" title="E74">A44</font></i></a>, <a class="txt" href="lattice5.html#E7:81_9"><i><font color="Green" title="E75">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="E10:81_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="lattice5.html#E8:81_9"><i><font color="Green" title="E76">A46</font></i></a></i>;<br/>
<a NAME="E11:81_9"/><b>then </b><i><font color="Green" title="E77">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="E12:81_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="lattice5.html#E11:81_9"><i><font color="Green" title="E77">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" target="_self" title="LATTICE5:def.7">Def7</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E27:81"/><i><font color="Green" title="E73">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="81_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" target="_self" 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="E2:81_10"/><i><font color="Green" title="E74">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="lattice5.html#E1:81"><i><font color="Green" title="E61">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="E4:81_10"/><i><font color="Green" title="E75">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="lattice5.html#E2:81_10"><i><font color="Green" title="E74">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="E6:81_10"/>
<font color="Maroon" title="c21">Aq1</font>,<font color="Maroon" title="c22">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c20">d1</font>
 <b>and </b><br/><a NAME="E7:81_10"/><i><font color="Green" title="E76">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E8:81_10"/><i><font color="Green" title="E77">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="lattice5.html#E7:81_10"><i><font color="Green" title="E76">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="E9:81_10"/><i><font color="Green" title="E78">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="lattice5.html#E7:81_10"><i><font color="Green" title="E76">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="E10:81_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="lattice5.html#E8:81_10"><i><font color="Green" title="E77">A52</font></i></a></i>;<br/>
<a NAME="E11:81_10"/><b>then </b><i><font color="Green" title="E79">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="E12:81_10"/><i><font color="Green" title="E80">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="lattice5.html#E2:81_10"><i><font color="Green" title="E74">A49</font></i></a>, <a class="txt" href="lattice5.html#E4:81_10"><i><font color="Green" title="E75">A50</font></i></a>, <a class="txt" href="lattice5.html#E7:81_10"><i><font color="Green" title="E76">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="E14:81_10"/><i><font color="Green" title="E81">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="lattice5.html#E1:81"><i><font color="Green" title="E61">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="E16:81_10"/><i><font color="Green" title="E82">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="lattice5.html#E14:81_10"><i><font color="Green" title="E81">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="E18:81_10"/>
<font color="Maroon" title="c27">Aq2</font>,<font color="Maroon" title="c28">dq2</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c25">A2</font>,<font color="Maroon" title="c26">d2</font>
 <b>and </b><br/><a NAME="E19:81_10"/><i><font color="Green" title="E83">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E20:81_10"/><i><font color="Green" title="E84">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="lattice5.html#E19:81_10"><i><font color="Green" title="E83">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="E21:81_10"/><i><font color="Green" title="E85">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="lattice5.html#E19:81_10"><i><font color="Green" title="E83">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="E22:81_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="lattice5.html#E20:81_10"><i><font color="Green" title="E84">A59</font></i></a></i>;<br/>
<a NAME="E23:81_10"/><b>then </b><i><font color="Green" title="E86">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="E24:81_10"/><i><font color="Green" title="E87">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="lattice5.html#E14:81_10"><i><font color="Green" title="E81">A56</font></i></a>, <a class="txt" href="lattice5.html#E16:81_10"><i><font color="Green" title="E82">A57</font></i></a>, <a class="txt" href="lattice5.html#E19:81_10"><i><font color="Green" title="E83">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="81_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_10_1"/>
( <font color="Maroon" 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="81_10_1_1"><b>suppose </b></a><a NAME="E1:81_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="E2:81_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="lattice5.html#E9:81_10"><i><font color="Green" title="E78">A53</font></i></a>, <a class="txt" href="lattice5.html#E21:81_10"><i><font color="Green" title="E85">A60</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</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="lattice5.html#E12:81_10"><i><font color="Green" title="E80">A55</font></i></a>, <a class="txt" href="lattice5.html#E14:81_10"><i><font color="Green" title="E81">A56</font></i></a>, <a class="txt" href="lattice5.html#E16:81_10"><i><font color="Green" title="E82">A57</font></i></a>, <a class="txt" href="lattice5.html#E19:81_10"><i><font color="Green" title="E83">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="E4:81_10_1_1"/><i><font color="Green" title="E88">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="lattice5.html#E23:81_10"><i><font color="Green" title="E86">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" target="_self" title="LATTICE5:def.6">Def6</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="lattice5.html#E23:81_10"><i><font color="Green" title="E86">A61</font></i></a>, <a class="txt" href="lattice5.html#E4:81_10_1_1"><i><font color="Green" title="E88">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="81_10_1_2"><b>suppose </b></a><a NAME="E1:81_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="E2:81_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="lattice5.html#E9:81_10"><i><font color="Green" title="E78">A53</font></i></a>, <a class="txt" href="lattice5.html#E21:81_10"><i><font color="Green" title="E85">A60</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</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="lattice5.html#E2:81_10"><i><font color="Green" title="E74">A49</font></i></a>, <a class="txt" href="lattice5.html#E4:81_10"><i><font color="Green" title="E75">A50</font></i></a>, <a class="txt" href="lattice5.html#E7:81_10"><i><font color="Green" title="E76">A51</font></i></a>, <a class="txt" href="lattice5.html#E24:81_10"><i><font color="Green" title="E87">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="E4:81_10_1_2"/><i><font color="Green" title="E88">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="lattice5.html#E11:81_10"><i><font color="Green" title="E79">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" target="_self" title="LATTICE5:def.6">Def6</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="lattice5.html#E11:81_10"><i><font color="Green" title="E79">A54</font></i></a>, <a class="txt" href="lattice5.html#E4:81_10_1_2"><i><font color="Green" title="E88">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="E28:81"/>
<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="81_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" target="_self" 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="E2:81_11"/><i><font color="Green" title="E74">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="lattice5.html#E1:81"><i><font color="Green" title="E61">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="E4:81_11"/><i><font color="Green" title="E75">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="lattice5.html#E2:81_11"><i><font color="Green" title="E74">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="E6:81_11"/>
<font color="Maroon" title="c22">Aq1</font>,<font color="Maroon" title="c23">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c20">A1</font>,<font color="Maroon" title="c21">d1</font>
 <b>and </b><br/><a NAME="E7:81_11"/><i><font color="Green" title="E76">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E8:81_11"/><i><font color="Green" title="E77">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="lattice5.html#E7:81_11"><i><font color="Green" title="E76">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="E9:81_11"/><i><font color="Green" title="E78">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="lattice5.html#E7:81_11"><i><font color="Green" title="E76">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="E10:81_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="lattice5.html#E8:81_11"><i><font color="Green" title="E77">A68</font></i></a></i>;<br/>
<a NAME="E11:81_11"/><b>then </b><i><font color="Green" title="E79">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="E12:81_11"/><i><font color="Green" title="E80">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="lattice5.html#E2:81_11"><i><font color="Green" title="E74">A65</font></i></a>, <a class="txt" href="lattice5.html#E4:81_11"><i><font color="Green" title="E75">A66</font></i></a>, <a class="txt" href="lattice5.html#E7:81_11"><i><font color="Green" title="E76">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="E13:81_11"/><i><font color="Green" title="E81">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="E15:81_11"/><i><font color="Green" title="E82">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="lattice5.html#E1:81"><i><font color="Green" title="E61">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="E17:81_11"/><i><font color="Green" title="E83">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="lattice5.html#E15:81_11"><i><font color="Green" title="E82">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="E19:81_11"/>
<font color="Maroon" title="c28">Aq2</font>,<font color="Maroon" title="c29">dq2</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c26">A2</font>,<font color="Maroon" title="c27">d2</font>
 <b>and </b><br/><a NAME="E20:81_11"/><i><font color="Green" title="E84">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E21:81_11"/><i><font color="Green" title="E85">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="lattice5.html#E20:81_11"><i><font color="Green" title="E84">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="E22:81_11"/><i><font color="Green" title="E86">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="lattice5.html#E20:81_11"><i><font color="Green" title="E84">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="E23:81_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="lattice5.html#E21:81_11"><i><font color="Green" title="E85">A76</font></i></a></i>;<br/>
<a NAME="E24:81_11"/><b>then </b><i><font color="Green" title="E87">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="E25:81_11"/><i><font color="Green" title="E88">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="lattice5.html#E15:81_11"><i><font color="Green" title="E82">A73</font></i></a>, <a class="txt" href="lattice5.html#E17:81_11"><i><font color="Green" title="E83">A74</font></i></a>, <a class="txt" href="lattice5.html#E20:81_11"><i><font color="Green" title="E84">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="E26:81_11"/><i><font color="Green" title="E89">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="E28:81_11"/><i><font color="Green" title="E90">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="lattice5.html#E1:81"><i><font color="Green" title="E61">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="E30:81_11"/><i><font color="Green" title="E91">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="lattice5.html#E28:81_11"><i><font color="Green" title="E90">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="E32:81_11"/>
<font color="Maroon" title="c34">Aq3</font>,<font color="Maroon" title="c35">dq3</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c32">A3</font>,<font color="Maroon" title="c33">d3</font>
 <b>and </b><br/><a NAME="E33:81_11"/><i><font color="Green" title="E92">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#K1" title="NAT_1:func.1">+</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="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E34:81_11"/><i><font color="Green" title="E93">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="lattice5.html#E33:81_11"><i><font color="Green" title="E92">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="E35:81_11"/><i><font color="Green" title="E94">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="lattice5.html#E33:81_11"><i><font color="Green" title="E92">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="E36:81_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="lattice5.html#E34:81_11"><i><font color="Green" title="E93">A84</font></i></a></i>;<br/>
<a NAME="E37:81_11"/><b>then </b><i><font color="Green" title="E95">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="E38:81_11"/><i><font color="Green" title="E96">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="lattice5.html#E28:81_11"><i><font color="Green" title="E90">A81</font></i></a>, <a class="txt" href="lattice5.html#E30:81_11"><i><font color="Green" title="E91">A82</font></i></a>, <a class="txt" href="lattice5.html#E33:81_11"><i><font color="Green" title="E92">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="E39:81_11"/><i><font color="Green" title="E97">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="81_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_11_1"/>
( <font color="Maroon" 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="81_11_1_1"><b>suppose </b></a><a NAME="E1:81_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="E2:81_11_1_1"/><b>then </b><i><font color="Green" title="E98">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="lattice5.html#E9:81_11"><i><font color="Green" title="E78">A69</font></i></a>, <a class="txt" href="lattice5.html#E22:81_11"><i><font color="Green" title="E86">A77</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</a></i>;<br/><b>thus </b><a NAME="E3:81_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="81_11_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_11_1_1_1_1"/>
( <font color="Maroon" 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="81_11_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:81_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="E2:81_11_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E99">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="lattice5.html#E22:81_11"><i><font color="Green" title="E86">A77</font></i></a>, <a class="txt" href="lattice5.html#E35:81_11"><i><font color="Green" title="E94">A85</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</a></i>;<br/><a NAME="E3:81_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="lattice5.html#E2:81_11_1_1"><i><font color="Green" title="E98">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="lattice5.html#E12:81_11"><i><font color="Green" title="E80">A71</font></i></a>, <a class="txt" href="lattice5.html#E25:81_11"><i><font color="Green" title="E88">A79</font></i></a>, <a class="txt" href="lattice5.html#E28:81_11"><i><font color="Green" title="E90">A81</font></i></a>, <a class="txt" href="lattice5.html#E30:81_11"><i><font color="Green" title="E91">A82</font></i></a>, <a class="txt" href="lattice5.html#E33:81_11"><i><font color="Green" title="E92">A83</font></i></a>, <a class="txt" href="lattice5.html#E2:81_11_1_1_1_1_1"><i><font color="Green" title="E99">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="E100">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="lattice5.html#E37:81_11"><i><font color="Green" title="E95">A86</font></i></a>, <a class="txt" href="lattice5.html#E39:81_11"><i><font color="Green" title="E97">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="E101">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="lattice5.html#E37:81_11"><i><font color="Green" title="E95">A86</font></i></a>, <a class="txt" href="lattice5.html#E39:81_11"><i><font color="Green" title="E97">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="lattice5.html#E37:81_11"><i><font color="Green" title="E95">A86</font></i></a>, <a class="txt" href="lattice5.html#E39:81_11"><i><font color="Green" title="E97">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="E8:81_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="lattice5.html#E5:81_11_1_1_1_1_1"><i><font color="Green" title="E100">A91</font></i></a>, <a class="txt" href="lattice5.html#E6:81_11_1_1_1_1_1"><i><font color="Green" title="E101">A92</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:81_11_1_1_1_1_2"/>
<font color="Maroon" 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="E2:81_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="lattice5.html#E22:81_11"><i><font color="Green" title="E86">A77</font></i></a>, <a class="txt" href="lattice5.html#E35:81_11"><i><font color="Green" title="E94">A85</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</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="lattice5.html#E12:81_11"><i><font color="Green" title="E80">A71</font></i></a>, <a class="txt" href="lattice5.html#E15:81_11"><i><font color="Green" title="E82">A73</font></i></a>, <a class="txt" href="lattice5.html#E17:81_11"><i><font color="Green" title="E83">A74</font></i></a>, <a class="txt" href="lattice5.html#E20:81_11"><i><font color="Green" title="E84">A75</font></i></a>, <a class="txt" href="lattice5.html#E38:81_11"><i><font color="Green" title="E96">A87</font></i></a>, <a class="txt" href="lattice5.html#E2:81_11_1_1"><i><font color="Green" title="E98">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="E99">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="lattice5.html#E24:81_11"><i><font color="Green" title="E87">A78</font></i></a>, <a class="txt" href="lattice5.html#E26:81_11"><i><font color="Green" title="E89">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="E100">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="lattice5.html#E24:81_11"><i><font color="Green" title="E87">A78</font></i></a>, <a class="txt" href="lattice5.html#E26:81_11"><i><font color="Green" title="E89">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="lattice5.html#E24:81_11"><i><font color="Green" title="E87">A78</font></i></a>, <a class="txt" href="lattice5.html#E26:81_11"><i><font color="Green" title="E89">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="E7:81_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="lattice5.html#E4:81_11_1_1_1_1_2"><i><font color="Green" title="E99">A93</font></i></a>, <a class="txt" href="lattice5.html#E5:81_11_1_1_1_1_2"><i><font color="Green" title="E100">A94</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_2"><b>suppose </b></a><a NAME="E1:81_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="E2:81_11_1_2"/><b>then </b><i><font color="Green" title="E98">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="lattice5.html#E9:81_11"><i><font color="Green" title="E78">A69</font></i></a>, <a class="txt" href="lattice5.html#E22:81_11"><i><font color="Green" title="E86">A77</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</a></i>;<br/><b>thus </b><a NAME="E3:81_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="81_11_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_11_1_2_1_1"/>
( <font color="Maroon" 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="81_11_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:81_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="E2:81_11_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E99">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="lattice5.html#E9:81_11"><i><font color="Green" title="E78">A69</font></i></a>, <a class="txt" href="lattice5.html#E35:81_11"><i><font color="Green" title="E94">A85</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</a></i>;<br/><a NAME="E3:81_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="lattice5.html#E2:81_11_1_2"><i><font color="Green" title="E98">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="lattice5.html#E12:81_11"><i><font color="Green" title="E80">A71</font></i></a>, <a class="txt" href="lattice5.html#E25:81_11"><i><font color="Green" title="E88">A79</font></i></a>, <a class="txt" href="lattice5.html#E28:81_11"><i><font color="Green" title="E90">A81</font></i></a>, <a class="txt" href="lattice5.html#E30:81_11"><i><font color="Green" title="E91">A82</font></i></a>, <a class="txt" href="lattice5.html#E33:81_11"><i><font color="Green" title="E92">A83</font></i></a>, <a class="txt" href="lattice5.html#E2:81_11_1_2_1_1_1"><i><font color="Green" title="E99">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="E100">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="lattice5.html#E37:81_11"><i><font color="Green" title="E95">A86</font></i></a>, <a class="txt" href="lattice5.html#E39:81_11"><i><font color="Green" title="E97">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="E101">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="lattice5.html#E37:81_11"><i><font color="Green" title="E95">A86</font></i></a>, <a class="txt" href="lattice5.html#E39:81_11"><i><font color="Green" title="E97">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="lattice5.html#E37:81_11"><i><font color="Green" title="E95">A86</font></i></a>, <a class="txt" href="lattice5.html#E39:81_11"><i><font color="Green" title="E97">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="E8:81_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="lattice5.html#E5:81_11_1_2_1_1_1"><i><font color="Green" title="E100">A97</font></i></a>, <a class="txt" href="lattice5.html#E6:81_11_1_2_1_1_1"><i><font color="Green" title="E101">A98</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_11_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:81_11_1_2_1_1_2"/>
<font color="Maroon" 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="E2:81_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="lattice5.html#E9:81_11"><i><font color="Green" title="E78">A69</font></i></a>, <a class="txt" href="lattice5.html#E35:81_11"><i><font color="Green" title="E94">A85</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</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="lattice5.html#E2:81_11"><i><font color="Green" title="E74">A65</font></i></a>, <a class="txt" href="lattice5.html#E4:81_11"><i><font color="Green" title="E75">A66</font></i></a>, <a class="txt" href="lattice5.html#E7:81_11"><i><font color="Green" title="E76">A67</font></i></a>, <a class="txt" href="lattice5.html#E25:81_11"><i><font color="Green" title="E88">A79</font></i></a>, <a class="txt" href="lattice5.html#E38:81_11"><i><font color="Green" title="E96">A87</font></i></a>, <a class="txt" href="lattice5.html#E2:81_11_1_2"><i><font color="Green" title="E98">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="E99">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="lattice5.html#E11:81_11"><i><font color="Green" title="E79">A70</font></i></a>, <a class="txt" href="lattice5.html#E13:81_11"><i><font color="Green" title="E81">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="E100">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="lattice5.html#E11:81_11"><i><font color="Green" title="E79">A70</font></i></a>, <a class="txt" href="lattice5.html#E13:81_11"><i><font color="Green" title="E81">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="lattice5.html#E11:81_11"><i><font color="Green" title="E79">A70</font></i></a>, <a class="txt" href="lattice5.html#E13:81_11"><i><font color="Green" title="E81">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="E7:81_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="lattice5.html#E4:81_11_1_2_1_1_2"><i><font color="Green" title="E99">A99</font></i></a>, <a class="txt" href="lattice5.html#E5:81_11_1_2_1_1_2"><i><font color="Green" title="E100">A100</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<b>hence </b><a NAME="E29:81"/>
 <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="lattice5.html#E26:81"><i><font color="Green" title="E72">A42</font></i></a>, <a class="txt" href="lattice5.html#E27:81"><i><font color="Green" title="E73">A48</font></i></a></i>;<br/>


</div>
