<?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>;<br/>

<b>assume </b><a NAME="E1:65"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/>

<b>given </b><font color="Maroon" title="c3">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">S</font><b> such that </b><a NAME="E3:65"/><i><font color="Green" title="E43">A2</font></i>: 
( <font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> &amp; <font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 <b>and </b><br/><a NAME="E4:65"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c3">g</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">S'</font> = <font color="Maroon" title="c2">S</font> as  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel_5.html#E3:65"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="waybel_5.html#E4:65"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="waybel_5.html#T29" target="_self" title="WAYBEL_5:th.29">Th29</a></i>;<br/>
<a NAME="E6:65"/>
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> <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><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="c4">S'</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">J</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c6">K</font> be  <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="c5">J</font>;<br/><b>let </b><font color="Maroon" title="c7">F</font> be   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c4">S'</font>;<br/>





<b>assume </b><a NAME="E1:65_1"/><i><font color="Green" title="E45">A4</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="c5">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="c7">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>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">d</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:65_1"/><i><font color="Green" title="E46">A5</font></i>: 
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c8">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>
 <b>and </b><br/><a NAME="E4:65_1"/>
for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> holds  <font color="Maroon" title="c8">d</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:65"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="waybel_1.html#T15" title="WAYBEL_1:th.15">WAYBEL_1:15</a></i>;<br/>
<a NAME="E5:65_1"/>
for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> holds  <font color="Maroon" title="c8">d</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">t</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#E4:65"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="waybel_5.html#E3:65_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="waybel_1.html#T23" title="WAYBEL_1:th.23">WAYBEL_1:23</a></i>;<br/>
<a NAME="E6:65_1"/><b>then </b><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">d</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T24" title="WAYBEL_1:th.24">WAYBEL_1:24</a></i>;<br/>
<a NAME="E7:65_1"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c8">d</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:65_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="waybel_1.html#T9" title="WAYBEL_1:th.9">WAYBEL_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">dF</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c7">F</font> as   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel_5.html#T31" target="_self" title="WAYBEL_5:th.31">Th31</a></i>;<br/>
<a NAME="E9:65_1"/><i><font color="Green" title="E49">A8</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="c5">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="c9">dF</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="65_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">J</font>;<br/>

<a NAME="E1:65_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c8">d</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>then </b><i><font color="Green" title="E50">A9</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="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon" title="c8">d</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T160" title="RELAT_1:th.160">RELAT_1:160</a></i>
;<br/>
<a NAME="E3:65_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c10">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#E1:65_1"><i><font color="Green" title="E45">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:65_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c10">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#E7:65_1"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="waybel_5.html#E2:65_1_1"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="yellow_2.html#T17" title="YELLOW_2:th.17">YELLOW_2:17</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c10">gdF</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c9">dF</font> as   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c2">S</font> ;<br/>
<i><font color="Green" title="E50">A10</font></i>: <font color="Maroon" title="c7">F</font> = 
<span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c2">S</font></span>)</span></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T4" title="MSUALG_3:th.4">MSUALG_3:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E6:65_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="waybel_5.html#T30" target="_self" title="WAYBEL_5:th.30">Th30</a></i>
<br/>.= 
<font color="Maroon" title="c10">gdF</font>
<b>by </b><i><a class="ref" href="autalg_1.html#T13" title="AUTALG_1:th.13">AUTALG_1:13</a></i>
;<br/>
<a NAME="E12:65_1"/><i><font color="Green" title="E51">A11</font></i>: 
 <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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:65_1_3"/>
<font color="Maroon" title="c11">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"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">J</font><b> such that </b><br/><a NAME="E3:65_1_3"/><i><font color="Green" title="E52">A12</font></i>: 
<font color="Maroon" title="c11">y</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/>
<font color="Maroon" title="c10">gdF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E5:65_1_3"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">gdF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T160" title="RELAT_1:th.160">RELAT_1:160</a></i>;<br/>
<a NAME="E6:65_1_3"/><b>then </b><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c11">y</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"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:65_1_3"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E7:65_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">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#E9:65_1"><i><font color="Green" title="E49">A8</font></i></a></i>;<br/>
<a NAME="E8:65_1_3"/><b>then </b><i><font color="Green" title="E54">A14</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:65"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E9:65_1_3"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E10:65_1_3"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E8:65_1_3"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E11:65_1_3"/><b>then </b><i><font color="Green" title="E55">A15</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E6:65_1_3"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E12:65_1_3"/>
(  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <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> &amp;  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <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/>
<b>hence </b><a NAME="E13:65_1_3"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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="txt" href="waybel_5.html#E11:65_1_3"><i><font color="Green" title="E55">A15</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:65_1"/><i><font color="Green" title="E52">A16</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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="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"><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="65_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:65_1_4"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:65_1_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E4:65_1_4"/><i><font color="Green" title="E53">A17</font></i>: 
<font color="Maroon" title="c12">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>and </b><br/><a NAME="E5:65_1_4"/><i><font color="Green" title="E54">A18</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">J</font><b> such that </b><br/><a NAME="E7:65_1_4"/><i><font color="Green" title="E55">A19</font></i>: 
<font color="Maroon" title="c12">x</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="txt" href="waybel_5.html#E4:65_1_4"><i><font color="Green" title="E53">A17</font></i></a>, <a class="ref" href="waybel_5.html#T14" target="_self" title="WAYBEL_5:th.14">Th14</a></i>;<br/>
<a NAME="E8:65_1_4"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">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#E9:65_1"><i><font color="Green" title="E49">A8</font></i></a></i>;<br/>
<a NAME="E9:65_1_4"/><b>then </b><i><font color="Green" title="E56">A20</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:65"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E10:65_1_4"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E11:65_1_4"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E9:65_1_4"><i><font color="Green" title="E56">A20</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E12:65_1_4"/><b>then </b><i><font color="Green" title="E57">A21</font></i>: 
<font color="Maroon" title="c11">y</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"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E5:65_1_4"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="waybel_5.html#E7:65_1_4"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/>
<font color="Maroon" title="c10">gdF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E14:65_1_4"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">gdF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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="c9">dF</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T160" title="RELAT_1:th.160">RELAT_1:160</a></i>;<br/>
<a NAME="E15:65_1_4"/><b>then </b>
<font color="Maroon" title="c11">y</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="txt" href="waybel_5.html#E12:65_1_4"><i><font color="Green" title="E57">A21</font></i></a>, <a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E16:65_1_4"/>
<font color="Maroon" title="c11">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"><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/>


</div><b>end;</b></div>
<a NAME="E14:65_1"/><i><font color="Green" title="E53">A22</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</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>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E15:65_1"/><i><font color="Green" title="E54">A23</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</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="txt" href="waybel_5.html#E3:65"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E16:65_1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> &amp; not  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> 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#E9:65_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="waybel_5.html#T27" target="_self" title="WAYBEL_5:th.27">Th27</a></i>;<br/>
<a NAME="E17:65_1"/><b>then </b><i><font color="Green" title="E55">A24</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</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#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:65"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E18:65_1"/><i><font color="Green" title="E56">A25</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</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#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E19:65_1"/><i><font color="Green" title="E57">A26</font></i>: 
for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c9">dF</font></span>)</span> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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="c7">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:65_1_5"/><i><font color="Green" title="E58">A27</font></i>: 
<font color="Maroon" title="c11">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="c9">dF</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">f</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">dF</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green" title="E59">A28</font></i>:  <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="c9">dF</font></span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">dF</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">F</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_5.html#T32" target="_self" title="WAYBEL_5:th.32">Th32</a></i>
<br/>.= 
 <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="c7">F</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<i><font color="Green" title="E60">A29</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">dF</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:65_1_5"><i><font color="Green" title="E58">A27</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="c5">J</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<i><font color="Green" title="E61">A30</font></i>:  <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="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:65_1_5"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="waybel_5.html#E3:65_1_5"><i><font color="Green" title="E59">A28</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="c5">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="E6:65_1_5"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:65_1_5"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font>
 
;<br/>
<a NAME="E8:65_1_5"/><b>then </b><i><font color="Green" title="E62">A31</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_5_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:65_1_5_4"/><i><font color="Green" title="E63">A32</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c14">j</font> = <font color="Maroon" title="c13">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">J</font> ;<br/><a NAME="E3:65_1_5_4"/><i><font color="Green" title="E64">A33</font></i>: 
<font color="Maroon" title="c14">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="c9">dF</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:65_1_5_4"><i><font color="Green" title="E63">A32</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:65_1_5_4"/><b>then </b>
<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#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="c9">dF</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#E1:65_1_5"><i><font color="Green" title="E58">A27</font></i></a>, <a class="ref" href="waybel_5.html#T9" target="_self" title="WAYBEL_5:th.9">Th9</a></i>;<br/><a NAME="E5:65_1_5_4"/><b>then </b>
<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#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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>;<br/><a NAME="E6:65_1_5_4"/><b>then </b><i><font color="Green" title="E65">A34</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#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="c8">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E7:65_1_5_4"/><i><font color="Green" title="E66">A35</font></i>: 
<font color="Maroon" title="c14">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="c7">F</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:65_1_5_4"><i><font color="Green" title="E63">A32</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">j</font> = 
<font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E4:65_1_5"><i><font color="Green" title="E60">A29</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">dF</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> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:65_1_5"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="waybel_5.html#E3:65_1_5_4"><i><font color="Green" title="E64">A33</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="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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>
<b>by </b><i><a class="txt" href="waybel_5.html#E6:65_1_5_4"><i><font color="Green" title="E65">A34</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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>
<b>by </b><i><a class="txt" href="waybel_5.html#E6:65_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <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>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
<br/>.= 
<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="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</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="txt" href="waybel_5.html#E1:65_1_5"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="waybel_5.html#E3:65_1_5"><i><font color="Green" title="E59">A28</font></i></a>, <a class="txt" href="waybel_5.html#E7:65_1_5_4"><i><font color="Green" title="E66">A35</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="E9:65_1_5_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 ;<br/></div><b>end;</b></div>
<b>then </b> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E4:65_1_5"><i><font color="Green" title="E60">A29</font></i></a>, <a class="txt" href="waybel_5.html#E5:65_1_5"><i><font color="Green" title="E61">A30</font></i></a>, <a class="txt" href="waybel_5.html#E8:65_1_5"><i><font color="Green" title="E62">A31</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T160" title="RELAT_1:th.160">RELAT_1:160</a></i>
;<br/>
<b>hence </b><a NAME="E11:65_1_5"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">f</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E20:65_1"/><i><font color="Green" title="E58">A36</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span></span>)</span> <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"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:65_1_6"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:65_1_6"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E4:65_1_6"/><i><font color="Green" title="E59">A37</font></i>: 
<font color="Maroon" title="c12">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#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 <b>and </b><br/><a NAME="E5:65_1_6"/><i><font color="Green" title="E60">A38</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:65_1_6"/><i><font color="Green" title="E61">A39</font></i>: 
<font color="Maroon" title="c13">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="c9">dF</font></span>)</span>
 <b>and </b><br/><a NAME="E8:65_1_6"/><i><font color="Green" title="E62">A40</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c9">dF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:65_1_6"><i><font color="Green" title="E59">A37</font></i></a>, <a class="ref" href="waybel_5.html#T13" target="_self" title="WAYBEL_5:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c13">f</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">dF</font></span>)</span> <b>by </b><i><a class="txt" href="waybel_5.html#E7:65_1_6"><i><font color="Green" title="E61">A39</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">f'</font> = <font color="Maroon" title="c14">f</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">F</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_5.html#T32" target="_self" title="WAYBEL_5:th.32">Th32</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">X</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span>;<br/>
<a NAME="E11:65_1_6"/><i><font color="Green" title="E63">A41</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:65"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E12:65_1_6"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E13:65_1_6"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E11:65_1_6"><i><font color="Green" title="E63">A41</font></i></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>;<br/>
<a NAME="E14:65_1_6"/><b>then </b><i><font color="Green" title="E64">A42</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E5:65_1_6"><i><font color="Green" title="E60">A38</font></i></a>, <a class="txt" href="waybel_5.html#E8:65_1_6"><i><font color="Green" title="E62">A40</font></i></a>, <a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>;<br/>
<i><font color="Green" title="E65">A43</font></i>:  <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="c9">dF</font></span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">dF</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">F</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_5.html#T32" target="_self" title="WAYBEL_5:th.32">Th32</a></i>
<br/>.= 
 <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="c7">F</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E16:65_1_6"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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="c7">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E19:65_1"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="waybel_5.html#E7:65_1_6"><i><font color="Green" title="E61">A39</font></i></a></i>;<br/>
<a NAME="E17:65_1_6"/><b>then </b>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 
<b>by </b><i><a class="txt" href="waybel_5.html#E14:65_1_6"><i><font color="Green" title="E64">A42</font></i></a>, <a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E18:65_1_6"/>
<font color="Maroon" title="c11">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"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E7:65_1_6"><i><font color="Green" title="E61">A39</font></i></a>, <a class="txt" href="waybel_5.html#E15:65_1_6"><i><font color="Green" title="E65">A43</font></i></a>, <a class="ref" href="waybel_5.html#T13" target="_self" title="WAYBEL_5:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:65_1"/><i><font color="Green" title="E59">A44</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:65_1_7"/>
<font color="Maroon" title="c11">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"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:65_1_7"/><i><font color="Green" title="E60">A45</font></i>: 
<font color="Maroon" title="c12">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="c7">F</font></span>)</span>
 <b>and </b><br/><a NAME="E4:65_1_7"/><i><font color="Green" title="E61">A46</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font></span>)</span>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="waybel_5.html#T13" target="_self" title="WAYBEL_5:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c12">f</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">F</font></span>)</span> <b>by </b><i><a class="txt" href="waybel_5.html#E3:65_1_7"><i><font color="Green" title="E60">A45</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">f'</font> = <font color="Maroon" title="c13">f</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">dF</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_5.html#T32" target="_self" title="WAYBEL_5:th.32">Th32</a></i>;<br/>
<i><font color="Green" title="E62">A47</font></i>:  <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="c9">dF</font></span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">dF</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">F</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_5.html#T32" target="_self" title="WAYBEL_5:th.32">Th32</a></i>
<br/>.= 
 <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="c7">F</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E8:65_1_7"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="c7">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E19:65_1"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="waybel_5.html#E3:65_1_7"><i><font color="Green" title="E60">A45</font></i></a></i>;<br/>
<a NAME="E9:65_1_7"/><b>then </b><i><font color="Green" title="E63">A48</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">f</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E4:65_1_7"><i><font color="Green" title="E61">A46</font></i></a>, <a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">X</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f'</font></span>)</span>;<br/>
<a NAME="E10:65_1_7"/><i><font color="Green" title="E64">A49</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:65"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E11:65_1_7"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f'</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<b>then </b><i><font color="Green" title="E65">A50</font></i>: <font color="Maroon" title="c11">y</font> = 
<font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c9">dF</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c14">f'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E9:65_1_7"><i><font color="Green" title="E63">A48</font></i></a>, <a class="txt" href="waybel_5.html#E10:65_1_7"><i><font color="Green" title="E64">A49</font></i></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>
;<br/>
<a NAME="E13:65_1_7"/>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <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#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:65_1_7"><i><font color="Green" title="E60">A45</font></i></a>, <a class="txt" href="waybel_5.html#E7:65_1_7"><i><font color="Green" title="E62">A47</font></i></a>, <a class="ref" href="waybel_5.html#T13" target="_self" title="WAYBEL_5:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E14:65_1_7"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E12:65_1_7"><i><font color="Green" title="E65">A50</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</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="txt" href="waybel_5.html#E11:65_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>
<br/>.= 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E12:65_1"><i><font color="Green" title="E51">A11</font></i></a>, <a class="txt" href="waybel_5.html#E13:65_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E14:65_1"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="waybel_5.html#E15:65_1"><i><font color="Green" title="E54">A23</font></i></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> </span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:65"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="waybel_5.html#E9:65_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="txt" href="waybel_5.html#E27"><i><font color="Green" title="E22">Lm8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span></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">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E17:65_1"><i><font color="Green" title="E55">A24</font></i></a>, <a class="txt" href="waybel_5.html#E18:65_1"><i><font color="Green" title="E56">A25</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>
<br/>.= 
 <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"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E20:65_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="waybel_5.html#E21:65_1"><i><font color="Green" title="E59">A44</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E23:65_1"/>
 <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> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:65"/>
<font color="Maroon" title="c2">S</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel_5.html#E29"><i><font color="Green" title="E24">Lm9</font></i></a></i>;<br/>


</div>
