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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <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">c1</font>, <font color="Olive" title="b2">c2</font> being <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">c1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Olive" title="b2">c2</font> iff  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b2">c2</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">c1</font> )</span><br/><b>let </b><font color="Maroon" title="c2">c1</font>, <font color="Maroon" title="c3">c2</font> be  <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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"> ( <font color="Maroon" title="c2">c1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">c2</font> iff  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font> )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font> implies <font color="Maroon" title="c2">c1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">c2</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">c1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font></span><br/><a NAME="E2:29_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:29_1_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T12" title="YELLOW_2:th.12">YELLOW_2:12</a></i>;<br/>
<a NAME="E4:29_1_1"/>
( <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> &amp; <font color="Maroon" title="c2">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> &amp; <font color="Maroon" title="c2">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> )
 
<b>by </b><i><a class="txt" href="#E1:29_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="waybel10.html#T6" target="_self" title="WAYBEL10:th.6">Th6</a>, <a class="ref" href="yellow_2.html#T10" title="YELLOW_2:th.10">YELLOW_2:10</a>, <a class="ref" href="yellow_2.html#T20" title="YELLOW_2:th.20">YELLOW_2:20</a></i>;<br/>
<a NAME="E5:29_1_1"/><b>then </b>
( <font color="Maroon" title="c2">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="#E3:29_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>
<a NAME="E6:29_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">c1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E7:29_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</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><b>hence </b><a NAME="E3:29_1"/>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font>
 <b>by </b><i><a class="ref" href="waybel10.html#T14" target="_self" title="WAYBEL10:th.14">Th14</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>

<b>assume </b><a NAME="E2:29"/><i><font color="Green" title="E16">A3</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font></span>)</span> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c1</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">c1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">c2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span><br/><a NAME="E1:29_2"/>
<font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">c2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/><a NAME="E2:29_2"/><b>then </b>
<font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c3">c2</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><a NAME="E3:29_2"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c2">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="yellow_2.html#T12" title="YELLOW_2:th.12">YELLOW_2:12</a></i>;<br/><a NAME="E4:29_2"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c2">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="ref" href="waybel10.html#T6" target="_self" title="WAYBEL10:th.6">Th6</a>, <a class="ref" href="yellow_2.html#T20" title="YELLOW_2:th.20">YELLOW_2:20</a></i>;<br/><b>hence </b><a NAME="E5:29_2"/>
<font color="Maroon" title="c2">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</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>
<b>hence </b><a NAME="E4:29"/>
<font color="Maroon" title="c2">c1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">c2</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T10" title="YELLOW_2:th.10">YELLOW_2:10</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>
