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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel33.html#V1" title="WAYBEL33:attr.1">lim-inf</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff for <font color="Olive" title="b2">F</font> being <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">F</font> holds <br/> <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff for <font color="Olive" title="b1">F</font> being <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/>



<a NAME="E1:32"/><i><font color="Green" title="E23">A1</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel33.html#D2" target="_self" title="WAYBEL33:def.2">Def2</a></i>;<br/>
<a NAME="E2:32"/>
 <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel28.html#D4" title="WAYBEL28:def.4">WAYBEL28:def 4</a></i>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green" title="E24">A2</font></i>: 
 <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : for <font color="Olive" title="b2">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> holds <br/>for <font color="Olive" title="b3">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">N</font>,<font color="Olive" title="b2">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b3">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27" title="YELLOW_6:def.27">YELLOW_6:def 27</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">F</font> being <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> ) implies <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/><a NAME="E2:32_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><a NAME="E3:32_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:32_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>and </b><br/><a NAME="E6:32_1"/><i><font color="Green" title="E26">A4</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/>for <font color="Olive" title="b2">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">N</font>,<font color="Olive" title="b1">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be  <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> implies  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/><a NAME="E7:32_1"/><i><font color="Green" title="E27">A5</font></i>: 
for <font color="Olive" title="b1">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of  <a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font> holds   <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">M</font>
 <b>by </b><i><a class="ref" href="waybel33.html#T16" target="_self" title="WAYBEL33:th.16">Th16</a></i>;<br/><a NAME="E8:32_1"/>
(  <a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel33.html#T13" target="_self" title="WAYBEL33:th.13">Th13</a>, <a class="ref" href="waybel33.html#T14" target="_self" title="WAYBEL33:th.14">Th14</a></i>;<br/><a NAME="E9:32_1"/><b>then </b><i><font color="Green" title="E28">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c4">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E7:32_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="waybel28.html#D3" title="WAYBEL28:def.3">WAYBEL28:def 3</a></i>;<br/><b>assume </b><a NAME="E10:32_1"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/><b>assume </b><a NAME="E11:32_1"/>
not  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E12:32_1"/><b>then </b>
 <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E13:32_1"/><b>then </b>
 <a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E5:32_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E6:32_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="#E9:32_1"><i><font color="Green" title="E28">A6</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span><b> such that </b><br/><a NAME="E15:32_1"/><i><font color="Green" title="E30">A8</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span>  st <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a></i>;<br/><a NAME="E16:32_1"/><i><font color="Green" title="E31">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</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"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Olive" title="b2">f</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">F</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">f</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow19.html#D4" title="YELLOW19:def.4">YELLOW19:def 4</a></i>;<br/><a NAME="E17:32_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c7">f</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">F</font><b> such that </b><br/><a NAME="E19:32_1"/><i><font color="Green" title="E32">A10</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> )
 <b>by </b><i><a class="txt" href="#E16:32_1"><i><font color="Green" title="E31">A9</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">A'</font> = <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c9">f'</font> = <font color="Maroon" title="c7">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E10:32_1"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">B</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E22:32_1"/><i><font color="Green" title="E33">A11</font></i>: 
( <font color="Maroon" title="c10">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c8">A'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c9">f'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">B</font> )
 <b>by </b><i><a class="txt" href="#E10:32_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">B</font>;<br/><a NAME="E24:32_1"/>
not  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T8" title="WAYBEL_7:th.8">WAYBEL_7:8</a></i>;<br/><a NAME="E25:32_1"/><b>then </b>
not <font color="Maroon" title="c10">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="#E22:32_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="yellow_1.html#T18" title="YELLOW_1:th.18">YELLOW_1:18</a></i>;<br/><a NAME="E26:32_1"/><b>then </b><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 ;<br/><a NAME="E27:32_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T4" title="WAYBEL_7:th.4">WAYBEL_7:4</a></i>;<br/><a NAME="E28:32_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c10">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:32_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="#E22:32_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="#E26:32_1"><i><font color="Green" title="E34">A12</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">j</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c10">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span> ;<br/><a NAME="E30:32_1"/><i><font color="Green" title="E35">A13</font></i>: 
( <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="#E22:32_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/><a NAME="E31:32_1"/><b>then </b>
<font color="Maroon" title="c12">j</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E32:32_1"/><b>then </b>
<font color="Maroon" title="c12">j</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">i</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="txt" href="#E19:32_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E33:32_1"/><b>then </b>
<font color="Maroon" title="c12">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="yellow19.html#D4" title="YELLOW19:def.4">YELLOW19:def 4</a></i>;<br/><a NAME="E34:32_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E15:32_1"><i><font color="Green" title="E30">A8</font></i></a></i>;<br/><a NAME="E35:32_1"/><b>then </b>
<font color="Maroon" title="c12">j</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="yellow19.html#D4" title="YELLOW19:def.4">YELLOW19:def 4</a></i>;<br/><a NAME="E36:32_1"/><b>then </b>
( <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="#E26:32_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="#E30:32_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E37:32_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:32"/><i><font color="Green" title="E25">A14</font></i>: 
for <font color="Olive" title="b1">F</font> being <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<b>set </b><font color="Maroon" title="c3">V</font> = <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">N</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">N</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/><b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><b>let </b><font color="Maroon" title="c6">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">N</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c6">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><b>assume </b><a NAME="E3:32_2"/><i><font color="Green" title="E27">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">N</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/><b>assume </b><a NAME="E4:32_2"/>
not <font color="Maroon" title="c6">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E5:32_2"/><b>then </b>
<font color="Maroon" title="c6">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T10" title="WAYBEL_0:th.10">WAYBEL_0:10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">N'</font> being  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a>  <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c6">N</font><b> such that </b><br/><a NAME="E7:32_2"/><i><font color="Green" title="E28">A17</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">N'</font> is    <a href="yellow_6.html#M1" title="YELLOW_6:mode.1">SubNetStr</a> of <font color="Maroon" title="c6">N</font> )
 <b>by </b><i><a class="ref" href="waybel33.html#T17" target="_self" title="WAYBEL33:th.17">Th17</a></i>;<br/><a NAME="E8:32_2"/>
 <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#D21" title="YELLOW_6:def.21">YELLOW_6:def 21</a></i>;<br/><a NAME="E9:32_2"/><b>then </b><i><font color="Green" title="E29">A18</font></i>: 
<font color="Maroon" title="c6">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E3:32_2"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:32_2"/><i><font color="Green" title="E30">A19</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">N</font>
 <b>by </b><i><a class="txt" href="#E7:32_2"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="yellow_6.html#T19" title="YELLOW_6:th.19">YELLOW_6:19</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">N1</font> being  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E12:32_2"/><i><font color="Green" title="E31">A20</font></i>: 
<font color="Maroon" title="c8">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N</font>
 <b>and </b><br/><a NAME="E13:32_2"/><i><font color="Green" title="E32">A21</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E9:32_2"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></i>;<br/><a NAME="E14:32_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E10:32_2"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="#E12:32_2"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="#E13:32_2"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/><a NAME="E15:32_2"/><b>then </b><i><font color="Green" title="E33">A22</font></i>: 
<font color="Maroon" title="c7">N'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></i>;<br/><a NAME="E16:32_2"/><i><font color="Green" title="E34">A23</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c7">N'</font>
 <b>by </b><i><a class="txt" href="#E3:32_2"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="#E9:32_2"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="waybel28.html#D3" title="WAYBEL28:def.3">WAYBEL28:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">M</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c7">N'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c9">M</font></span><br/><a NAME="E1:32_2_1"/>
<font color="Maroon" title="c9">M</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c6">N</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T24" title="YELLOW_6:th.24">YELLOW_6:24</a></i>;<br/><b>hence </b><a NAME="E2:32_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c9">M</font>
 <b>by </b><i><a class="txt" href="#E3:32_2"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="#E9:32_2"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="waybel28.html#D3" title="WAYBEL28:def.3">WAYBEL28:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E18:32_2"/><b>then </b><i><font color="Green" title="E35">A24</font></i>: 
for <font color="Olive" title="b1">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c7">N'</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">M</font>
 ;<br/><b>set </b><font color="Maroon" title="c9">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c10">j2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font>;<br/><b>set </b><font color="Maroon" title="c11">g</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c10">j2</font></span>)</span>;<br/><a NAME="E20:32_2"/><i><font color="Green" title="E36">A25</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c10">j2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span> 
 ;<br/><a NAME="E21:32_2"/>
for <font color="Olive" title="b1">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b2">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span>  holds <br/><font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span>  implies <font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:32_2_2"/>
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c13">j3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font><b> such that </b><br/><a NAME="E3:32_2_2"/><i><font color="Green" title="E37">A26</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">j3</font></span>)</span>
 ;<br/>
<a NAME="E4:32_2_2"/>
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:32_2_2"><i><font color="Green" title="E37">A26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:32_2_2"/>
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_7.html#T4" title="WAYBEL_7:th.4">WAYBEL_7:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c12">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E20:32_2"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E23:32_2"/><i><font color="Green" title="E37">A27</font></i>: 
not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_3"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c13">Y</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c12">G</font><b> such that </b><br/><a NAME="E3:32_2_3"/><i><font color="Green" title="E38">A28</font></i>: 
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c13">Y</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c13">Y</font>, <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">Y</font> = <font color="Maroon" title="c13">Y</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<i><font color="Green" title="E39">A29</font></i>:  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> = 
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D12" title="YELLOW_0:def.12">YELLOW_0:def 12</a></i>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T19" title="YELLOW_1:th.19">YELLOW_1:19</a></i>
;<br/>
<b>defpred </b><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> ] <b>means </b> ex <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span> );<br/>
<a NAME="E6:32_2_3"/><i><font color="Green" title="E40">A30</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N'</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N'</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:32_2_3_2"/><i><font color="Green" title="E41">A31</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N'</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>assume </b><a NAME="E2:32_2_3_2"/><i><font color="Green" title="E42">A32</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N'</font> holds <br/> not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Olive" title="b1">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:32_2_3_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">G</font>
 
<b>by </b><i><a class="txt" href="#E1:32_2_3_2"><i><font color="Green" title="E41">A31</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font><b> such that </b><br/><a NAME="E5:32_2_3_2"/><i><font color="Green" title="E43">A33</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c16">j</font></span>)</span>
 ;<br/>
<b>thus </b><a NAME="E6:32_2_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:32_2_3_2"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="#E5:32_2_3_2"><i><font color="Green" title="E43">A33</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:32_2_3"/><i><font color="Green" title="E41">A34</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Y</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N'</font> )
 <b>and </b><br/><a NAME="E9:32_2_3"/><i><font color="Green" title="E42">A35</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="#E6:32_2_3"><i><font color="Green" title="E40">A30</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c16">C</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">N'</font></span>)</span> <b>by </b><i><a class="txt" href="#E8:32_2_3"><i><font color="Green" title="E41">A34</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>
<a NAME="E11:32_2_3"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">N'</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D6" title="WAYBEL_0:def.6">WAYBEL_0:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">j0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font><b> such that </b><br/><a NAME="E13:32_2_3"/><i><font color="Green" title="E43">A36</font></i>: 
( <font color="Maroon" title="c17">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">N'</font> &amp; <font color="Maroon" title="c17">j0</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c16">C</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></i>;<br/>
<a NAME="E14:32_2_3"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> holds <br/> <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> implies  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">y</font> )</span><br/>

<b>assume </b><a NAME="E1:32_2_3_3"/><i><font color="Green" title="E44">A37</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">y</font></span><br/>

<a NAME="E2:32_2_3_3"/><i><font color="Green" title="E45">A38</font></i>: 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f</font>
 
<b>by </b><i><a class="txt" href="#E8:32_2_3"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="#E1:32_2_3_3"><i><font color="Green" title="E44">A37</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">f1</font> = <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> <b>by </b><i><a class="txt" href="#E8:32_2_3"><i><font color="Green" title="E41">A34</font></i></a></i>;<br/>
<a NAME="E4:32_2_3_3"/><i><font color="Green" title="E46">A39</font></i>: 
<font color="Maroon" title="c19">f1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c17">j0</font>
 
<b>by </b><i><a class="txt" href="#E13:32_2_3"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="#E2:32_2_3_3"><i><font color="Green" title="E45">A38</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">j1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font><b> such that </b><br/><a NAME="E6:32_2_3_3"/><i><font color="Green" title="E47">A40</font></i>: 
( <font color="Maroon" title="c20">j1</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> <font color="Maroon" title="c18">y</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c20">j1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E9:32_2_3"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="#E1:32_2_3_3"><i><font color="Green" title="E44">A37</font></i></a></i>;<br/>
<a NAME="E7:32_2_3_3"/>
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span> holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span> implies <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">y</font> )</span><br/>

<b>assume </b><a NAME="E1:32_2_3_3_1"/><i><font color="Green" title="E48">A41</font></i>: 
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">y</font></span><br/>

<a NAME="E2:32_2_3_3_1"/>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:32_2_3_3_1"><i><font color="Green" title="E48">A41</font></i></a>, <a class="ref" href="waybel_9.html#D7" title="WAYBEL_9:def.7">WAYBEL_9:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:32_2_3_3_1"/><i><font color="Green" title="E49">A42</font></i>: 
( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">q</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:32_2_3_3_1"/>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:32_2_3_3_1"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:32_2_3_3_1"/><b>then </b><i><font color="Green" title="E50">A43</font></i>: 
( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> &amp; <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:32_2_3_3_1"/><b>then </b>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n'</font> where <font color="Olive" title="b1">n'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : <font color="Maroon" title="c17">j0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">n'</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_9.html#T12" title="WAYBEL_9:th.12">WAYBEL_9:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">n'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font><b> such that </b><br/><a NAME="E9:32_2_3_3_1"/><i><font color="Green" title="E51">A44</font></i>: 
( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n'</font> &amp; <font color="Maroon" title="c17">j0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">n'</font> )
 ;<br/>
<a NAME="E10:32_2_3_3_1"/><i><font color="Green" title="E52">A45</font></i>: 
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">q</font>
 
<b>by </b><i><a class="txt" href="#E4:32_2_3_3_1"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E11:32_2_3_3_1"/>
<font color="Maroon" title="c19">f1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">n'</font>
 
<b>by </b><i><a class="txt" href="#E4:32_2_3_3"><i><font color="Green" title="E46">A39</font></i></a>, <a class="txt" href="#E9:32_2_3_3_1"><i><font color="Green" title="E51">A44</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E12:32_2_3_3_1"/><b>then </b>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">m'</font> where <font color="Olive" title="b1">m'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : <font color="Maroon" title="c20">j1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">m'</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E6:32_2_3_3"><i><font color="Green" title="E47">A40</font></i></a>, <a class="txt" href="#E9:32_2_3_3_1"><i><font color="Green" title="E51">A44</font></i></a></i>;<br/>
<a NAME="E13:32_2_3_3_1"/><b>then </b>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c20">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T12" title="WAYBEL_9:th.12">WAYBEL_9:12</a></i>;<br/>
<a NAME="E14:32_2_3_3_1"/><b>then </b>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c20">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:32_2_3_3_1"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E15:32_2_3_3_1"/><b>then </b><i><font color="Green" title="E53">A46</font></i>: 
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c20">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E16:32_2_3_3_1"/><b>then </b>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c20">j1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">q</font>
 
<b>by </b><i><a class="txt" href="#E10:32_2_3_3_1"><i><font color="Green" title="E52">A45</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E17:32_2_3_3_1"/><b>then </b>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c20">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E15:32_2_3_3_1"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E18:32_2_3_3_1"/>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">y</font>
 <b>by </b><i><a class="txt" href="#E6:32_2_3_3"><i><font color="Green" title="E47">A40</font></i></a>, <a class="ref" href="waybel_9.html#D7" title="WAYBEL_9:def.7">WAYBEL_9:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_2_3_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:32_2_3"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">Y</font>
 
<b>by </b><i><a class="txt" href="#E3:32_2_3"><i><font color="Green" title="E38">A28</font></i></a>, <a class="txt" href="#E5:32_2_3"><i><font color="Green" title="E39">A29</font></i></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/>
<a NAME="E16:32_2_3"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c17">j0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E3:32_2_3"><i><font color="Green" title="E38">A28</font></i></a>, <a class="txt" href="#E5:32_2_3"><i><font color="Green" title="E39">A29</font></i></a>, <a class="ref" href="yellow_1.html#T20" title="YELLOW_1:th.20">YELLOW_1:20</a></i>;<br/>
<b>hence </b><a NAME="E17:32_2_3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E24:32_2"/>
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">y</font> holds <br/> not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">y</font> implies  not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font> )</span><br/>

<b>assume </b><a NAME="E1:32_2_4"/><i><font color="Green" title="E38">A47</font></i>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font></span><br/>

<a NAME="E2:32_2_4"/><i><font color="Green" title="E39">A48</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T18" title="YELLOW_1:th.18">YELLOW_1:18</a></i>;<br/>
<a NAME="E3:32_2_4"/>
<font color="Maroon" title="c13">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:32_2_4"><i><font color="Green" title="E38">A47</font></i></a>, <a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:32_2_4"/>
not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font>
 <b>by </b><i><a class="txt" href="#E23:32_2"><i><font color="Green" title="E37">A27</font></i></a>, <a class="txt" href="#E2:32_2_4"><i><font color="Green" title="E39">A48</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E25:32_2"/><b>then </b>
not  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></i>;<br/><a NAME="E26:32_2"/><b>then </b>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font></span>)</span> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>
 <b>by </b><i><a class="ref" href="waybel_7.html#T8" title="WAYBEL_7:th.8">WAYBEL_7:8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">F</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E28:32_2"/><i><font color="Green" title="E38">A49</font></i>: 
(  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">F</font> &amp; <font color="Maroon" title="c13">F</font> is <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T30" title="WAYBEL_7:th.30">WAYBEL_7:30</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">j0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font>;<br/><a NAME="E30:32_2"/><i><font color="Green" title="E39">A50</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c14">j0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">G</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c15">rj</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c14">j0</font></span>)</span>, <font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel_7.html#T4" title="WAYBEL_7:th.4">WAYBEL_7:4</a></i>;<br/><a NAME="E32:32_2"/><i><font color="Green" title="E40">A51</font></i>: 
( <font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font> &amp;  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a>, <a class="ref" href="waybel_0.html#T50" title="WAYBEL_0:th.50">WAYBEL_0:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rj</font> implies <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> )</span><br/><b>assume </b><a NAME="E1:32_2_5"/><i><font color="Green" title="E41">A52</font></i>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rj</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font></span><br/><a NAME="E2:32_2_5"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c14">j0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:32_2_5"><i><font color="Green" title="E41">A52</font></i></a>, <a class="ref" href="waybel_9.html#D7" title="WAYBEL_9:def.7">WAYBEL_9:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:32_2_5"/><i><font color="Green" title="E42">A53</font></i>: 
( <font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c14">j0</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c14">j0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">q</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:32_2_5"/>
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c14">j0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:32_2_5"><i><font color="Green" title="E42">A53</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E6:32_2_5"/><b>then </b><i><font color="Green" title="E43">A54</font></i>: 
( <font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> &amp; <font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c14">j0</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:32_2_5"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">q</font>
 <b>by </b><i><a class="txt" href="#E4:32_2_5"><i><font color="Green" title="E42">A53</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E8:32_2_5"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font>
 <b>by </b><i><a class="txt" href="#E6:32_2_5"><i><font color="Green" title="E43">A54</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E34:32_2"/><b>then </b>
<font color="Maroon" title="c15">rj</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E35:32_2"/><b>then </b>
<font color="Maroon" title="c15">rj</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">a</font>
 <b>by </b><i><a class="txt" href="#E7:32_2"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E36:32_2"/><b>then </b>
<font color="Maroon" title="c15">rj</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">a</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/><a NAME="E37:32_2"/><b>then </b><i><font color="Green" title="E41">A55</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E30:32_2"><i><font color="Green" title="E39">A50</font></i></a>, <a class="txt" href="#E32:32_2"><i><font color="Green" title="E40">A51</font></i></a>, <a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></i>;<br/><a NAME="E38:32_2"/><i><font color="Green" title="E42">A56</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span> ,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E16:32_2"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="waybel33.html#T11" target="_self" title="WAYBEL33:th.11">Th11</a></i>;<br/><a NAME="E39:32_2"/><i><font color="Green" title="E43">A57</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span> ,<font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/><a NAME="E40:32_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span>  or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:32_2_6"/><i><font color="Green" title="E44">A58</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font> </span>}</span> </span><br/>

<b>consider </b><font color="Maroon" title="c18">j3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font><b> such that </b><br/><a NAME="E3:32_2_6"/><i><font color="Green" title="E45">A59</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c18">j3</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:32_2_6"><i><font color="Green" title="E44">A58</font></i></a></i>;<br/>
<i><font color="Green" title="E46">A60</font></i>: <font color="Maroon" title="c17">x</font> = 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
<b>by </b><i><a class="txt" href="#E3:32_2_6"><i><font color="Green" title="E45">A59</font></i></a>, <a class="ref" href="waybel_9.html#D2" title="WAYBEL_9:def.2">WAYBEL_9:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c18">j3</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c19">f1</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c18">j3</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E6:32_2_6"/>
 <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">F</font>
 
<b>by </b><i><a class="txt" href="#E28:32_2"><i><font color="Green" title="E38">A49</font></i></a>, <a class="txt" href="#E32:32_2"><i><font color="Green" title="E40">A51</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:32_2_6"/><b>then </b>
( <font color="Maroon" title="c19">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">G</font> &amp; <font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">F</font> )
 
<b>by </b><i><a class="txt" href="#E32:32_2"><i><font color="Green" title="E40">A51</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E8:32_2_6"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:32_2_6"><i><font color="Green" title="E46">A60</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E41:32_2"/><b>then </b><i><font color="Green" title="E44">A61</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c13">F</font>
 <b>by </b><i><a class="txt" href="#E38:32_2"><i><font color="Green" title="E42">A56</font></i></a>, <a class="txt" href="#E39:32_2"><i><font color="Green" title="E43">A57</font></i></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></i>;<br/><a NAME="E42:32_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font> </span>}</span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font> </span>}</span>  or <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<b>assume </b><a NAME="E1:32_2_7"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c18">f0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:32_2_7"/><i><font color="Green" title="E45">A62</font></i>: 
( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c18">f0</font> &amp; <font color="Maroon" title="c18">f0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font> )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font>] <b>means </b>( $1 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> $2 &amp; <font color="Maroon" title="c7">N'</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f0</font> );<br/>
<div><i><font color="Green" title="E46">A63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">pj</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">j</font>,<font color="Olive" title="b1">pj</font>]</span><br/><a NAME="E1:32_2_7_1"/>
<font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E32:32_2"><i><font color="Green" title="E40">A51</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E2:32_2_7_1"/><b>then </b>
( <font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">F</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">G</font> )
 <b>by </b><i><a class="txt" href="#E28:32_2"><i><font color="Green" title="E38">A49</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:32_2_7_1"/><b>then </b><i><font color="Green" title="E47">A64</font></i>: 
<font color="Maroon" title="c18">f0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font>
 <b>by </b><i><a class="txt" href="#E3:32_2_7"><i><font color="Green" title="E45">A62</font></i></a>, <a class="ref" href="waybel_7.html#T13" title="WAYBEL_7:th.13">WAYBEL_7:13</a></i>;<br/><a NAME="E4:32_2_7_1"/>
<font color="Maroon" title="c13">F</font> is  <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E28:32_2"><i><font color="Green" title="E38">A49</font></i></a></i>;<br/><a NAME="E5:32_2_7_1"/><b>then </b>
not  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T8" title="WAYBEL_7:th.8">WAYBEL_7:8</a></i>;<br/><a NAME="E6:32_2_7_1"/><b>then </b>
not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T18" title="YELLOW_1:th.18">YELLOW_1:18</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">nj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E8:32_2_7_1"/><i><font color="Green" title="E48">A65</font></i>: 
<font color="Maroon" title="c20">nj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:32_2_7_1"><i><font color="Green" title="E47">A64</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E9:32_2_7_1"/>
( <font color="Maroon" title="c20">nj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f0</font> &amp; <font color="Maroon" title="c20">nj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:32_2_7_1"><i><font color="Green" title="E48">A65</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">pj'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:32_2_7_1"/><i><font color="Green" title="E49">A66</font></i>: 
( <font color="Maroon" title="c21">pj'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span> &amp; <font color="Maroon" title="c20">nj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">pj'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:32_2_7_1"/>
<font color="Maroon" title="c21">pj'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:32_2_7_1"><i><font color="Green" title="E49">A66</font></i></a></i>;<br/><a NAME="E13:32_2_7_1"/><b>then </b>
<font color="Maroon" title="c21">pj'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y'</font> where <font color="Olive" title="b1">y'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> : <font color="Maroon" title="c19">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">y'</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_9.html#T12" title="WAYBEL_9:th.12">WAYBEL_9:12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">pj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font><b> such that </b><br/><a NAME="E15:32_2_7_1"/><i><font color="Green" title="E50">A67</font></i>: 
( <font color="Maroon" title="c22">pj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">pj'</font> &amp; <font color="Maroon" title="c19">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">pj</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c23">pj'</font> = <font color="Maroon" title="c21">pj'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span> <b>by </b><i><a class="txt" href="#E11:32_2_7_1"><i><font color="Green" title="E49">A66</font></i></a></i>;<br/><i><font color="Green" title="E51">A68</font></i>: <font color="Maroon" title="c7">N'</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c22">pj</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c23">pj'</font>
<b>by </b><i><a class="txt" href="#E15:32_2_7_1"><i><font color="Green" title="E50">A67</font></i></a>, <a class="ref" href="waybel_9.html#T16" title="WAYBEL_9:th.16">WAYBEL_9:16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c23">pj'</font>

;<br/><b>take </b><font color="Maroon" title="c24">pj</font> = <font color="Maroon" title="c22">pj</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c24">pj</font>]</span><br/><b>thus </b><a NAME="E18:32_2_7_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c24">pj</font>]
 <b>by </b><i><a class="txt" href="#E8:32_2_7_1"><i><font color="Green" title="E48">A65</font></i></a>, <a class="txt" href="#E11:32_2_7_1"><i><font color="Green" title="E49">A66</font></i></a>, <a class="txt" href="#E15:32_2_7_1"><i><font color="Green" title="E50">A67</font></i></a>, <a class="txt" href="#E17:32_2_7_1"><i><font color="Green" title="E51">A68</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c19">p</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c7">N'</font>,<font color="Maroon" title="c7">N'</font><b> such that </b><br/><a NAME="E6:32_2_7"/><i><font color="Green" title="E47">A69</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c19">p</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S10" title="FUNCT_2:sch.10">FUNCT_2:sch 10</a>(<a class="txt" href="#E4:32_2_7"><i><font color="Green" title="E46">A63</font></i></a>);<br/></i>
<a NAME="E7:32_2_7"/><i><font color="Green" title="E48">A70</font></i>: 
for <font color="Olive" title="b1">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c7">N'</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Maroon" title="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <font color="Olive" title="b1">M</font>
 
<b>by </b><i><a class="txt" href="#E15:32_2"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="#E18:32_2"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="waybel28.html#T3" title="WAYBEL28:th.3">WAYBEL28:3</a></i>;<br/>
<a NAME="E8:32_2_7"/>
<font color="Maroon" title="c19">p</font> is <a href="waybel28.html#V1" title="WAYBEL28:attr.1">greater_or_equal_to_id</a>
 
<b>by </b><i><a class="txt" href="#E6:32_2_7"><i><font color="Green" title="E47">A69</font></i></a>, <a class="ref" href="waybel28.html#D1" title="WAYBEL28:def.1">WAYBEL28:def 1</a></i>;<br/>
<a NAME="E9:32_2_7"/><b>then </b><i><font color="Green" title="E49">A71</font></i>: 
 <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="#E15:32_2"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="#E16:32_2"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="#E7:32_2_7"><i><font color="Green" title="E48">A70</font></i></a>, <a class="ref" href="waybel28.html#T14" title="WAYBEL28:th.14">WAYBEL28:14</a></i>;<br/>
<a NAME="E10:32_2_7"/>
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> holds <br/><font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> implies <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f0</font> )</span><br/>

<b>assume </b><a NAME="E1:32_2_7_2"/>
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f0</font></span><br/>

<a NAME="E2:32_2_7_2"/><b>then </b>
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel11.html#T19" title="WAYBEL11:th.19">WAYBEL11:19</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span><b> such that </b><br/><a NAME="E4:32_2_7_2"/><i><font color="Green" title="E50">A72</font></i>: 
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c21">k</font>
 ;<br/>
<a NAME="E5:32_2_7_2"/><i><font color="Green" title="E51">A73</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N'</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#T7" title="WAYBEL28:th.7">WAYBEL28:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">l</font> = <font color="Maroon" title="c21">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N'</font> <b>by </b><i><a class="ref" href="waybel28.html#T7" title="WAYBEL28:th.7">WAYBEL28:7</a></i>;<br/>
<a NAME="E7:32_2_7_2"/>
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N'</font>
 
<b>by </b><i><a class="txt" href="#E5:32_2_7_2"><i><font color="Green" title="E51">A73</font></i></a></i>;<br/>
<a NAME="E8:32_2_7_2"/><b>then </b><i><font color="Green" title="E52">A74</font></i>: 
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">p</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c21">k</font> = 
<span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font>
<b>by </b><i><a class="ref" href="waybel28.html#D2" title="WAYBEL28:def.2">WAYBEL28:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c7">N'</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c22">l</font></span>)</span>
<b>by </b><i><a class="txt" href="#E8:32_2_7_2"><i><font color="Green" title="E52">A74</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E10:32_2_7_2"/>
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">f0</font>
 <b>by </b><i><a class="txt" href="#E6:32_2_7"><i><font color="Green" title="E47">A69</font></i></a>, <a class="txt" href="#E4:32_2_7_2"><i><font color="Green" title="E50">A72</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:32_2_7"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">f0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:32_2_7"/><b>then </b><i><font color="Green" title="E50">A75</font></i>: 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c18">f0</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T3" title="WAYBEL_7:th.3">WAYBEL_7:3</a></i>;<br/>
 <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span> = 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
<b>by </b><i><a class="ref" href="waybel_9.html#D2" title="WAYBEL_9:def.2">WAYBEL_9:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c7">N'</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c19">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E14:32_2_7"/>
<font color="Maroon" title="c17">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="#E3:32_2_7"><i><font color="Green" title="E45">A62</font></i></a>, <a class="txt" href="#E9:32_2_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="txt" href="#E12:32_2_7"><i><font color="Green" title="E50">A75</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E43:32_2"/><b>then </b>
 <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c13">F</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/><a NAME="E44:32_2"/><b>then </b><i><font color="Green" title="E45">A76</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c13">F</font>
 <b>by </b><i><a class="txt" href="#E41:32_2"><i><font color="Green" title="E44">A61</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/><a NAME="E45:32_2"/>
not <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E1:32_2"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E46:32_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:32"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="#E28:32_2"><i><font color="Green" title="E38">A49</font></i></a>, <a class="txt" href="#E37:32_2"><i><font color="Green" title="E41">A55</font></i></a>, <a class="txt" href="#E44:32_2"><i><font color="Green" title="E45">A76</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:32"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<a NAME="E8:32"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:32"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="waybel12.html#D1" title="WAYBEL12:def.1">WAYBEL12:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
