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

<b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c2">f1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">y</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c2">f1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> ) ) ) &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c3">f2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">y</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c3">f2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> ) ) ) implies <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:76_1_2"/><i><font color="Green" title="E55">A14</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c2">f1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">y</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c2">f1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> ) )
 <b>and </b><br/><a NAME="E2:76_1_2"/><i><font color="Green" title="E56">A15</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c3">f2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">y</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c3">f2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font></span><br/>

<i><font color="Green" title="E57">A16</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</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 href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font>
<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="E4:76_1_2"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font> holds <br/><font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font> implies <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> )</span><br/>

<b>assume </b><a NAME="E1:76_1_2_2"/><i><font color="Green" title="E58">A17</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:76_1_2_2"/><i><font color="Green" title="E59">A18</font></i>: 
<font color="Maroon" title="c4">z</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="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="lattice5.html#E1:76_1_2_2"><i><font color="Green" title="E58">A17</font></i></a>, <a class="txt" href="lattice5.html#E3:76_1_2_2"><i><font color="Green" title="E59">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_2_2_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:76_1_2_2_1_1"/><i><font color="Green" title="E60">A19</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font></span><br/><div class="add"><b>thus </b><font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> = 
<font color="Maroon" title="c2">f1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_2_2"><i><font color="Green" title="E59">A18</font></i></a></i>
<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="txt" href="lattice5.html#E1:76_1_2"><i><font color="Green" title="E55">A14</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_2_2_1_1"><i><font color="Green" title="E60">A19</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>
<b>by </b><i><a class="txt" href="lattice5.html#E2:76_1_2"><i><font color="Green" title="E56">A15</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_2_2_1_1"><i><font color="Green" title="E60">A19</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_2_2"><i><font color="Green" title="E59">A18</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:76_1_2_2_1_2"/><i><font color="Green" title="E60">A20</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font></span><br/><div class="add"><b>thus </b><font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> = 
<font color="Maroon" title="c2">f1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_2_2"><i><font color="Green" title="E59">A18</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">y</font>
<b>by </b><i><a class="txt" href="lattice5.html#E1:76_1_2"><i><font color="Green" title="E55">A14</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_2_2_1_2"><i><font color="Green" title="E60">A20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>
<b>by </b><i><a class="txt" href="lattice5.html#E2:76_1_2"><i><font color="Green" title="E56">A15</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_2_2_1_2"><i><font color="Green" title="E60">A20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_2_2"><i><font color="Green" title="E59">A18</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:76_1_2"/>
<font color="Maroon" title="c2">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">f2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_2"><i><font color="Green" title="E57">A16</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
