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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a>  iff 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">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.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="c1">L</font> holds   <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:58"/>
( <font color="Maroon" title="c1">L</font> is  <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a>  implies 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">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.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="c1">L</font> holds   <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> <br/> for <font color="Olive" title="b2">K</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.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="c1">L</font> holds   <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  ) implies <font color="Maroon" title="c1">L</font> is  <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1"/>
<font color="Maroon" title="c1">L</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> 
 <span class="kw">and </span><br/><a NAME="E2:58_1"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
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">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.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="c1">L</font> holds   <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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_5.html#D3" title="WAYBEL_5:def.3">WAYBEL_5:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> <br/> for <font color="Olive" title="b2">K</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.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="c1">L</font> holds   <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c2">J</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">L</font> holds   <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c2">J</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">L</font> holds   <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span><br/>





<a NAME="E3:58_1"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1"><span class="lab"><font color="Green" title="E33">A1</font></span></a></span>;<br/>
<a NAME="E4:58_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E5:58_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T45" target="_self" title="YELLOW_7:th.45">Th45</a></span>;<br/>
<a NAME="E6:58_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E7:58_1"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T45" target="_self" title="YELLOW_7:th.45">Th45</a></span>;<br/>
<a NAME="E8:58_1"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 <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="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E9:58_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">J9</font> =  <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> <span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span></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/>
<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> or <font color="Maroon" title="c6">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">K9</font> = <font color="Maroon" title="c5">J9</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c5">J9</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">G</font> =  <a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> as   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c7">K9</font>,<font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">assume </span><a NAME="E4:58_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">J9</font><span class="kw"> such that </span><br/><a NAME="E6:58_1_1"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_5.html#T14" title="WAYBEL_5:th.14">WAYBEL_5:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">g9</font> = <font color="Maroon" title="c9">g</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1: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="c4">F</font></span>)</span> : <font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 </span>}</span> ;<br/>
<a NAME="E8:58_1_1"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>($1);<br/><span class="kw">assume </span><a NAME="E1:58_1_1_1"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
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="c2">J</font>  holds <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><span class="lab"><font color="Green" title="E43">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> implies  ex <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">j</font>,<font color="Olive" title="b1">k</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:58_1_1_1_1"/>
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">j</font>,<font color="Olive" title="b1">k</font>]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c11">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c13">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">j</font>);<br/><a NAME="E4:58_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">i</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1"><span class="lab"><font color="Green" title="E42">A10</font></span></a></span>;<br/><a NAME="E5:58_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">i</font>)
 ;<br/><span class="kw">hence </span><a NAME="E6:58_1_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">j</font>,<font color="Olive" title="b1">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c11">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:58_1_1_1"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> &amp; ( 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="c2">J</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E2:58_1_1_1"><span class="lab"><font color="Green" title="E43">A11</font></span></a>);<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> implies <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:58_1_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span><br/><a NAME="E2:58_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_1_1_2"/>
<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">h</font> = <font color="Maroon" title="c11">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1: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="c4">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E4:58_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><a NAME="E7:58_1_1_1"/>
<font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E9:58_1_1_1"/>
( <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">j</font>) &amp; <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:58_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font><span class="kw"> such that </span><br/><a NAME="E11:58_1_1"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>
<a NAME="E12:58_1_1"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> or <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:58_1_1_2"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:58_1_1_2"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">u</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a></span>;<br/>
<a NAME="E6:58_1_1_2"/>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1: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="c4">F</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:58_1_1_2"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:58_1_1_2"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
<font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font>
 ;<br/>
<a NAME="E10:58_1_1_2"/>
( <font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c10">g9</font> &amp; <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#D8" title="PRALG_2:def.8">PRALG_2:def 8</a></span>;<br/>
<a NAME="E11:58_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E9:58_1_1_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>;<br/>
<a NAME="E12:58_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E4:58_1_1_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E8:58_1_1_2"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E9:58_1_1_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>;<br/>
<a NAME="E13:58_1_1_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">K9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon" title="c7">K9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">F</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:58_1_1_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_1_1_2"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:58_1_1"/>
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>,<font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/>
<a NAME="E14:58_1_1"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <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#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_1_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="yellow_0.html#T35" title="YELLOW_0:th.35">YELLOW_0:35</a></span>;<br/>
<a NAME="E15:58_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></span>;<br/>
<a NAME="E16:58_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<font color="Maroon" title="c6">a</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></span>;<br/>
<a NAME="E17:58_1_1"/>
 <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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK7" title="WAYBEL_5:NK.7">Infs</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_5.html#T14" title="WAYBEL_5:th.14">WAYBEL_5:14</a></span>;<br/>
<a NAME="E18:58_1_1"/><span class="kw">then </span>
 <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>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK7" title="WAYBEL_5:NK.7">Infs</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></span>;<br/>
<a NAME="E19:58_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK7" title="WAYBEL_5:NK.7">Infs</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:58_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:58_1"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></span>;<br/>
<a NAME="E11:58_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A21</font></span>: 
 <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_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></span>;<br/>
<a NAME="E12:58_1"/>
 <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_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T47" target="_self" title="YELLOW_7:th.47">Th47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:58_1"/>
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E11:58_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:58"/><span class="lab"><font color="Green" title="E33">A22</font></span>: 
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">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.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="c1">L</font> holds   <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">L</font> is  <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:58"/>
<font color="Maroon" title="c1">L</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_5.html#D3" title="WAYBEL_5:def.3">WAYBEL_5:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive">b<sub>3</sub></font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c2">J</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c2">J</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">J</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font></span><br/>





<a NAME="E4:58"/><span class="lab"><font color="Green" title="E34">A23</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E5:58"/><span class="lab"><font color="Green" title="E35">A24</font></span>: 
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T45" target="_self" title="YELLOW_7:th.45">Th45</a></span>;<br/>
<a NAME="E6:58"/><span class="lab"><font color="Green" title="E36">A25</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E7:58"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T45" target="_self" title="YELLOW_7:th.45">Th45</a></span>;<br/>
<a NAME="E8:58"/><span class="lab"><font color="Green" title="E38">A27</font></span>: 
 <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="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E9:58"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK8" title="WAYBEL_5:NK.8">Sups</a> </span>)</span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">J9</font> =  <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> <span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span></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/>
<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK8" title="WAYBEL_5:NK.8">Sups</a> </span>)</span> or  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">K9</font> = <font color="Maroon" title="c5">J9</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c5">J9</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">G</font> =  <a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> as   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c7">K9</font>,<font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">assume </span><a NAME="E4:58_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK8" title="WAYBEL_5:NK.8">Sups</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">J9</font><span class="kw"> such that </span><br/><a NAME="E6:58_2"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_5.html#T14" title="WAYBEL_5:th.14">WAYBEL_5:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">g9</font> = <font color="Maroon" title="c9">g</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1: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="c4">F</font></span>)</span> : <font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 </span>}</span> ;<br/>
<a NAME="E8:58_2"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>($1);<br/><span class="kw">assume </span><a NAME="E1:58_2_1"/><span class="lab"><font color="Green" title="E41">A30</font></span>: 
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="c2">J</font>  holds <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><span class="lab"><font color="Green" title="E42">A31</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> implies  ex <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">j</font>,<font color="Olive" title="b1">k</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:58_2_1_1"/>
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">j</font>,<font color="Olive" title="b1">k</font>]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c11">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c13">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">j</font>);<br/><a NAME="E4:58_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">i</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_2_1"><span class="lab"><font color="Green" title="E41">A30</font></span></a></span>;<br/><a NAME="E5:58_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">i</font>)
 ;<br/><span class="kw">hence </span><a NAME="E6:58_2_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">j</font>,<font color="Olive" title="b1">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c11">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:58_2_1"/><span class="lab"><font color="Green" title="E43">A32</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> &amp; ( 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="c2">J</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E2:58_2_1"><span class="lab"><font color="Green" title="E42">A31</font></span></a>);<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> implies <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:58_2_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span><br/><a NAME="E2:58_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_2_1"><span class="lab"><font color="Green" title="E43">A32</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_2_1_2"/>
<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58"><span class="lab"><font color="Green" title="E37">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">h</font> = <font color="Maroon" title="c11">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1: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="c4">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="txt" href="#E4:58_2_1"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><a NAME="E7:58_2_1"/>
<font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="txt" href="#E8:58_2"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E9:58_2_1"/>
( <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">j</font>) &amp; <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_2_1"><span class="lab"><font color="Green" title="E43">A32</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:58_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font><span class="kw"> such that </span><br/><a NAME="E11:58_2"/><span class="lab"><font color="Green" title="E41">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>
<a NAME="E12:58_2"/><span class="lab"><font color="Green" title="E42">A34</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> or <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_2_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:58_2_2"/><span class="lab"><font color="Green" title="E43">A35</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:58_2_2"/><span class="lab"><font color="Green" title="E44">A36</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">u</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_2_2"><span class="lab"><font color="Green" title="E43">A35</font></span></a></span>;<br/>
<a NAME="E6:58_2_2"/>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_2"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1: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="c4">F</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:58_2_2"/><span class="lab"><font color="Green" title="E45">A37</font></span>: 
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:58_2_2"/><span class="lab"><font color="Green" title="E46">A38</font></span>: 
<font color="Maroon" title="c10">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font>
 ;<br/>
<a NAME="E10:58_2_2"/>
( <font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c10">g9</font> &amp; <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#D8" title="PRALG_2:def.8">PRALG_2:def 8</a></span>;<br/>
<a NAME="E11:58_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="txt" href="#E9:58_2_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>;<br/>
<a NAME="E12:58_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A39</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="txt" href="#E4:58_2_2"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="txt" href="#E8:58_2_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="txt" href="#E9:58_2_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>;<br/>
<a NAME="E13:58_2_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">K9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon" title="c7">K9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">F</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:58_2_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_2_2"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:58_2"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>,<font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/>
<a NAME="E14:58_2"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_2"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></span>;<br/>
<a NAME="E15:58_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_2"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></span>;<br/>
<a NAME="E16:58_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A40</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></span>;<br/>
<a NAME="E17:58_2"/>
 <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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_5.html#T14" title="WAYBEL_5:th.14">WAYBEL_5:14</a></span>;<br/>
<a NAME="E18:58_2"/><span class="kw">then </span>
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</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#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></span>;<br/>
<a NAME="E19:58_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</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#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_2"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:58_2"/>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:58"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK8" title="WAYBEL_5:NK.8">Sups</a> </span>)</span></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></span>;<br/>
<a NAME="E11:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A41</font></span>: 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></span>;<br/>
<a NAME="E12:58"/>
(  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  &amp;  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="ref" href="waybel_5.html#T16" title="WAYBEL_5:th.16">WAYBEL_5:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:58"/>
 <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
