<?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">J</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>   st <font color="Olive" title="b1">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds <br/>for <font color="Olive" title="b2">K</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V7</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Olive" title="b1">J</font>  st ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">J</font> holds  <font color="Olive" title="b2">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> ) holds <br/>for <font color="Olive" title="b3">F</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Olive" title="b2">K</font>,<font color="Maroon" title="c1">L</font>  st ( for <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">J</font> holds   <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b4">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) holds <br/> <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  ) implies <font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> )</span><br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E24">A1</font></i>: 
for <font color="Olive" title="b1">J</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>   st <font color="Olive" title="b1">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds <br/>for <font color="Olive" title="b2">K</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V7</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Olive" title="b1">J</font>  st ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">J</font> holds  <font color="Olive" title="b2">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> ) holds <br/>for <font color="Olive" title="b3">F</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Olive" title="b2">K</font>,<font color="Maroon" title="c1">L</font>  st ( for <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">J</font> holds   <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b4">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) holds <br/> <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span></span><br/><b>set </b><font color="Maroon" title="c3">J</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">j</font> </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c4">UN</font> =  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</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="E1:36_1"/><i><font color="Green" title="E25">A2</font></i>: 
 <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_6.html#D3" title="YELLOW_6:def.3">YELLOW_6:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">UN</font> =  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> as  <a href="classes2.html#V1" title="CLASSES2:attr.1">universal</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E3:36_1"/><i><font color="Green" title="E26">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</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="classes1.html#T59" title="CLASSES1:th.59">CLASSES1:59</a></i>;<br/><a NAME="E4:36_1"/>
 <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UN</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="classes1.html#T5" title="CLASSES1:th.5">CLASSES1:5</a></i>;<br/><a NAME="E5:36_1"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UN</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:36_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/><a NAME="E6:36_1"/><b>then </b><i><font color="Green" title="E28">A5</font></i>: 
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UN</font>
 <b>by </b><i><a class="ref" href="classes2.html#T65" title="CLASSES2:th.65">CLASSES2:65</a></i>;<br/><a NAME="E7:36_1"/><i><font color="Green" title="E29">A6</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">j</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> 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="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">u</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="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">j</font> </span>}</span>  or <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">j</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="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:36_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">j</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">j</font> )
 
;<br/>
<b>hence </b><a NAME="E3:36_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:36_1"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">j</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UN</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E6:36_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/><a NAME="E9:36_1"/><i><font color="Green" title="E31">A8</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T5" title="WAYBEL_0:th.5">WAYBEL_0:5</a></i>;<br/><a NAME="E10:36_1"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T39" title="YELLOW_0:th.39">YELLOW_0:39</a></i>;<br/><a NAME="E11:36_1"/><b>then </b><i><font color="Green" title="E32">A9</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">j</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E9:36_1"><i><font color="Green" title="E31">A8</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">J</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">j</font> </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/><a NAME="E13:36_1"/><i><font color="Green" title="E33">A10</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> holds  <font color="Olive" title="b1">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E1:36_1_2"/>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c8">j'</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:36_1_2"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c8">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 <b>and </b><br/><a NAME="E4:36_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c8">j'</font>
 ;<br/>
<b>thus </b><a NAME="E5:36_1_2"/>
<font color="Maroon" title="c7">j</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:36_1_2"><i><font color="Green" title="E34">A11</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><b>set </b><font color="Maroon" title="c7">K</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c6">J</font>;<br/><a NAME="E14:36_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c6">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">J</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">K</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c6">J</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c6">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E16:36_1"/><i><font color="Green" title="E34">A12</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> holds  <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UN</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UN</font></span><br/>

<a NAME="E1:36_1_3"/>
<font color="Maroon" title="c8">K</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> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<a NAME="E2:36_1_3"/><b>then </b>
<font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 
;<br/>
<b>hence </b><a NAME="E3:36_1_3"/>
<font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UN</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E5:36_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="waybel_5.html#E7:36_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</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><a NAME="E17:36_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> holds <br/> not <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</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="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> implies  not <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>

<b>assume </b><a NAME="E1:36_1_4"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c10">i'</font> = <font color="Maroon" title="c9">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> ;<br/>
<a NAME="E3:36_1_4"/>
<font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i'</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_4"/>
not <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="waybel_5.html#E13:36_1"><i><font color="Green" title="E33">A10</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><b>then reconsider </b><font color="Maroon" title="c9">K</font> = <font color="Maroon" title="c8">K</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V7</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c6">J</font> <b>by </b><i><a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,<font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/><a NAME="E19:36_1"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">J</font> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">j</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><b>then 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="E21:36_1"/><i><font color="Green" title="E35">A13</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">J</font>
 <b>and </b><br/><a NAME="E22:36_1"/><i><font color="Green" title="E36">A14</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> holds <br/><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c10">F</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c6">J</font> <b>by </b><i><a class="txt" href="waybel_5.html#E21:36_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E24:36_1"/>
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">F</font> holds <br/><font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">j</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="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">F</font> implies <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/>

<b>assume </b><a NAME="E1:36_1_5"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/>

<a NAME="E2:36_1_5"/><b>then </b>
<font color="Maroon" title="c11">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>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E21:36_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="waybel_5.html#E22:36_1"><i><font color="Green" title="E36">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:36_1_5"/>
<font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ; <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>then reconsider </b><font color="Maroon" title="c12">F</font> = <font color="Maroon" title="c11">F</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c6">J</font> <b>by </b><i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></i>;<br/><a NAME="E26:36_1"/>
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> holds <br/><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">j</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="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> implies <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> )</span><br/>

<b>assume </b><a NAME="E1:36_1_6"/>
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">j</font> = <font color="Maroon" title="c13">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> ;<br/>
<a NAME="E3:36_1_6"/><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E22:36_1"><i><font color="Green" title="E36">A14</font></i></a></i>;<br/>
<a NAME="E4:36_1_6"/><b>then </b><i><font color="Green" title="E38">A16</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</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="E5:36_1_6"/><i><font color="Green" title="E39">A17</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:36_1_6"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:36_1_6"/>
<font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<a NAME="E7:36_1_6"/><b>then </b>
<font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E13:36_1"><i><font color="Green" title="E33">A10</font></i></a></i>;<br/>
<a NAME="E8:36_1_6"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</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 <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E5:36_1_6"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:36_1_6"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E10:36_1_6"/>
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:36_1_6"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</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>then reconsider </b><font color="Maroon" title="c13">F</font> = <font color="Maroon" title="c12">F</font> as   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c9">K</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><a NAME="E28:36_1"/><i><font color="Green" title="E37">A18</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font><br/> for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</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">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font></span><br/><b>let </b><font color="Maroon" title="c15">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font></span><br/>



<a NAME="E1:36_1_7"/>
<font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E22:36_1"><i><font color="Green" title="E36">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:36_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</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><a NAME="E29:36_1"/><i><font color="Green" title="E38">A19</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> holds   <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">y</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="c15">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> implies <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">j</font> )</span><br/><b>assume </b><a NAME="E1:36_1_8_1"/>
<font color="Maroon" title="c15">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</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="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">j</font></span><br/><b>then consider </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:36_1_8_1"/><i><font color="Green" title="E39">A20</font></i>: 
<font color="Maroon" title="c16">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
 <b>and </b><br/><a NAME="E4:36_1_8_1"/><i><font color="Green" title="E40">A21</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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/><a NAME="E5:36_1_8_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:36_1_8_1"><i><font color="Green" title="E39">A20</font></i></a></i>;<br/><a NAME="E6:36_1_8_1"/><b>then </b>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E7:36_1_8_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E28:36_1"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="waybel_5.html#E3:36_1_8_1"><i><font color="Green" title="E39">A20</font></i></a>, <a class="txt" href="waybel_5.html#E4:36_1_8_1"><i><font color="Green" title="E40">A21</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>
<a NAME="E2:36_1_8"/><b>then </b><i><font color="Green" title="E39">A22</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</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="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">j</font> implies <font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:36_1_8_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span></span><br/><a NAME="E2:36_1_8_2"/><b>then </b><i><font color="Green" title="E40">A23</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E3:36_1_8_2"/><b>then </b><i><font color="Green" title="E41">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E28:36_1"><i><font color="Green" title="E37">A18</font></i></a></i>;<br/><a NAME="E4:36_1_8_2"/>
<font color="Maroon" title="c15">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E2:36_1_8_2"><i><font color="Green" title="E40">A23</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="E5:36_1_8_2"/>
<font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:36_1_8_2"><i><font color="Green" title="E41">A24</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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>
<a NAME="E4:36_1_8"/><b>then </b>
<font color="Maroon" title="c14">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_8"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E2:36_1_8"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 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><b>end;</b></div><a NAME="E30:36_1"/><i><font color="Green" title="E39">A25</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> holds   <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a></span><br/>

<a NAME="E1:36_1_9"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E29:36_1"><i><font color="Green" title="E38">A19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:36_1_9"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 <b>by </b><i><a class="txt" href="waybel_5.html#E13:36_1"><i><font color="Green" title="E33">A10</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><a NAME="E31:36_1"/>
for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> holds <br/> <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> implies  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:36_1_10"/><i><font color="Green" title="E40">A26</font></i>: 
<font color="Maroon" title="c14">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c15">a</font> =  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E2:36_1_10"/>
for <font color="Olive" title="b1">D</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">D</font> holds <br/> ex <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp;  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">d</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c16">D</font> implies  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">D</font> &amp;  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font> ) )</span><br/>

<b>assume </b><a NAME="E1:36_1_10_1"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c16">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">D</font> &amp;  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font> )</span><br/>

<a NAME="E2:36_1_10_1"/><b>then </b>
<font color="Maroon" title="c16">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">D'</font> = <font color="Maroon" title="c16">D</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> ;<br/>
<a NAME="E4:36_1_10_1"/><i><font color="Green" title="E41">A27</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> holds  <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span><br/>

<a NAME="E1:36_1_10_1_1"/>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 
;<br/>
<a NAME="E2:36_1_10_1_1"/><b>then </b>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">F</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:36_1_10_1_1"/><b>then </b>
<font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1_10"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="waybel_5.html#T9" target="_self" title="WAYBEL_5:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_10_1_1"/>
<font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 ; <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>
<i><font color="Green" title="E42">A28</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">f</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">F</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1_10"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="waybel_5.html#T8" target="_self" title="WAYBEL_5:th.8">Th8</a></i>
<br/>.= 
<font color="Maroon" title="c6">J</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">y</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="c18">y</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="c14">f</font> implies <font color="Maroon" title="c18">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> )</span><br/><b>assume </b><a NAME="E1:36_1_10_1_3"/>
<font color="Maroon" title="c18">y</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="c14">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">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></span><br/><b>then consider </b><font color="Maroon" title="c19">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:36_1_10_1_3"/><i><font color="Green" title="E43">A29</font></i>: 
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">f</font>
 <b>and </b><br/><a NAME="E4:36_1_10_1_3"/><i><font color="Green" title="E44">A30</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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/><b>reconsider </b><font color="Maroon" title="c20">j</font> = <font color="Maroon" title="c19">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> <b>by </b><i><a class="txt" href="waybel_5.html#E5:36_1_10_1"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="waybel_5.html#E3:36_1_10_1_3"><i><font color="Green" title="E43">A29</font></i></a></i>;<br/><a NAME="E6:36_1_10_1_3"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:36_1_10_1"><i><font color="Green" title="E41">A27</font></i></a>, <a class="txt" href="waybel_5.html#E4:36_1_10_1_3"><i><font color="Green" title="E44">A30</font></i></a></i>;<br/><a NAME="E7:36_1_10_1_3"/><b>then </b><i><font color="Green" title="E45">A31</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E8:36_1_10_1_3"/>
<font color="Maroon" title="c20">j</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E13:36_1"><i><font color="Green" title="E33">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E9:36_1_10_1_3"/>
<font color="Maroon" title="c18">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="waybel_5.html#E7:36_1_10_1_3"><i><font color="Green" title="E45">A31</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>
<a NAME="E7:36_1_10_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">f</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">f</font> = <font color="Maroon" title="c14">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">J</font>,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="waybel_5.html#E5:36_1_10_1"><i><font color="Green" title="E42">A28</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E9:36_1_10_1"/><i><font color="Green" title="E43">A32</font></i>: 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c17">D'</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T55" title="YELLOW_2:th.55">YELLOW_2:55</a></i>;<br/>
<i><font color="Green" title="E44">A33</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">F</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1_10"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="waybel_5.html#T8" target="_self" title="WAYBEL_5:th.8">Th8</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">f</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1_10"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="waybel_5.html#T8" target="_self" title="WAYBEL_5:th.8">Th8</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">j</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="c19">j</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="c18">f</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> )</span><br/><b>assume </b><a NAME="E1:36_1_10_1_5"/><i><font color="Green" title="E45">A34</font></i>: 
<font color="Maroon" title="c19">j</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="c18">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span><br/><a NAME="E2:36_1_10_1_5"/><b>then </b><i><font color="Green" title="E46">A35</font></i>: 
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">F</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1_10"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="waybel_5.html#T8" target="_self" title="WAYBEL_5:th.8">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">j'</font> = <font color="Maroon" title="c19">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> <b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1_10_1_5"><i><font color="Green" title="E45">A34</font></i></a></i>;<br/><a NAME="E4:36_1_10_1_5"/><i><font color="Green" title="E47">A36</font></i>: 
<font color="Maroon" title="c18">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c20">j'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j'</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:36_1_10_1"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c20">j'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c20">j'</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1_10"><i><font color="Green" title="E40">A26</font></i></a>, <a class="txt" href="waybel_5.html#E2:36_1_10_1_5"><i><font color="Green" title="E46">A35</font></i></a>, <a class="ref" href="waybel_5.html#T9" target="_self" title="WAYBEL_5:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E28:36_1"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="waybel_5.html#E4:36_1_10_1_5"><i><font color="Green" title="E47">A36</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>
<a NAME="E12:36_1_10_1"/><b>then </b><i><font color="Green" title="E45">A37</font></i>: 
 <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c17">D'</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E9:36_1_10_1"><i><font color="Green" title="E43">A32</font></i></a>, <a class="txt" href="waybel_5.html#E10:36_1_10_1"><i><font color="Green" title="E44">A33</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E13:36_1_10_1"/>
<font color="Maroon" title="c18">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c17">D'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">D'</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E4:36_1_10_1"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/>
<a NAME="E14:36_1_10_1"/><b>then </b>
<font color="Maroon" title="c18">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c17">D'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D'</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E15:36_1_10_1"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">D</font> &amp;  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E12:36_1_10_1"><i><font color="Green" title="E45">A37</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>
<a NAME="E3:36_1_10"/><b>then </b>
 <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D1" title="WAYBEL_3:def.1">WAYBEL_3:def 1</a></i>;<br/>
<a NAME="E4:36_1_10"/><b>then </b>
 <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T7" title="WAYBEL_3:th.7">WAYBEL_3:7</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_10"/>
 <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</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><a NAME="E32:36_1"/><b>then </b><i><font color="Green" title="E40">A38</font></i>: 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_5.html#T15" target="_self" title="WAYBEL_5:th.15">Th15</a></i>;<br/><a NAME="E33:36_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</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="c14">b</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> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span> or <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font> )</span><br/>

<b>assume </b><a NAME="E1:36_1_11"/>
<font color="Maroon" title="c14">b</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> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </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="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font></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">J</font><b> such that </b><br/><a NAME="E3:36_1_11"/><i><font color="Green" title="E41">A39</font></i>: 
<font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="ref" href="waybel_5.html#T14" target="_self" title="WAYBEL_5:th.14">Th14</a></i>;<br/>
<a NAME="E4:36_1_11"/>
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c16">j'</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:36_1_11"/><i><font color="Green" title="E42">A40</font></i>: 
<font color="Maroon" title="c16">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font>
 <b>and </b><br/><a NAME="E7:36_1_11"/><i><font color="Green" title="E43">A41</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c16">j'</font>
 ;<br/>
<font color="Maroon" title="c14">b</font> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E3:36_1_11"><i><font color="Green" title="E41">A39</font></i></a>, <a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c16">j'</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E29:36_1"><i><font color="Green" title="E38">A19</font></i></a>, <a class="txt" href="waybel_5.html#E6:36_1_11"><i><font color="Green" title="E42">A40</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:36_1_11"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E7:36_1_11"><i><font color="Green" title="E43">A41</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><a NAME="E34:36_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/><a NAME="E35:36_1"/><b>then </b><i><font color="Green" title="E41">A42</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 <b>by </b><i><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">j</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">J</font> <b>by </b><i><a class="txt" href="waybel_5.html#E11:36_1"><i><font color="Green" title="E32">A9</font></i></a></i>;<br/> <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="waybel_5.html#E29:36_1"><i><font color="Green" title="E38">A19</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T39" title="YELLOW_0:th.39">YELLOW_0:39</a></i>
;<br/><a NAME="E38:36_1"/><b>then </b>
<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel_5.html#T14" target="_self" title="WAYBEL_5:th.14">Th14</a></i>;<br/><a NAME="E39:36_1"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></i>;<br/><a NAME="E40:36_1"/><b>then </b>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>;<br/><b>then </b><i><font color="Green" title="E42">A43</font></i>: <font color="Maroon" title="c2">x</font> = 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
<b>by </b><i><a class="txt" href="waybel_5.html#E35:36_1"><i><font color="Green" title="E41">A42</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:36"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel_5.html#E8:36_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="waybel_5.html#E16:36_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="waybel_5.html#E30:36_1"><i><font color="Green" title="E39">A25</font></i></a></i>
;<br/><a NAME="E42:36_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T9" title="WAYBEL_3:th.9">WAYBEL_3:9</a></i>;<br/><a NAME="E43:36_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/><b>hence </b><a NAME="E44:36_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E32:36_1"><i><font color="Green" title="E40">A38</font></i></a>, <a class="txt" href="waybel_5.html#E41:36_1"><i><font color="Green" title="E42">A43</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</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>
<a NAME="E3:36"/><b>then </b><i><font color="Green" title="E25">A44</font></i>: 
( <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5" title="WAYBEL_3:def.5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E4:36"/>
for <font color="Olive" title="b1">x</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/> ( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
;<br/>
<b>hence </b><a NAME="E5:36"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:36"><i><font color="Green" title="E25">A44</font></i></a>, <a class="ref" href="waybel_3.html#D6" title="WAYBEL_3:def.6">WAYBEL_3:def 6</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>
