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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>  <a href="waybel19.html#NM1" title="WAYBEL19:NM.1">TopPoset</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="waybel32.html#V1" title="WAYBEL32:attr.1">upper</a>  implies <font color="Maroon" title="c1">T</font> is  <a href="waybel32.html#V2" title="WAYBEL32:attr.2">order_consistent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:7"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="waybel32.html#V1" title="WAYBEL32:attr.1">upper</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> is  <a href="waybel32.html#V2" title="WAYBEL32:attr.2">order_consistent</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">BB</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> : verum </span> } </span>  as   <a href="cantor_1.html#NM2" title="CANTOR_1:NM.2">prebasis</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:7"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> holds <br/> (  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; (  for <font color="Olive" title="b2">N</font> being   <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b2">N</font> holds <br/> for <font color="Olive" title="b3">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">x</font> holds  <font color="Olive" title="b2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b3">V</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; (  for <font color="Olive" title="b1">N</font> being   <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b1">N</font> holds <br/> for <font color="Olive" title="b2">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> holds  <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b2">V</font> ) )</span><br/>

<a NAME="E1:7_1"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">a9</font> = <font color="Maroon" title="c4">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:7_1_1"/>
 for <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">a9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">a9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">G</font> implies <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1_1_1"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c6">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">a9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">G</font> or <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:7_1_1_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c5">a9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">G</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c5">a9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E4:7_1_1_1"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/>
<a NAME="E5:7_1_1_1"/>
<font color="Maroon" title="c6">G</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:7_1_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="waybel32.html#T1" target="_self" onmouseover="rs('waybel32/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E6:7_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1_1_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E4:7_1_1_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a></span>;<br/>
<a NAME="E7:7_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:7_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_1_1_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:7_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T24" onmouseover="rs('pre_topc/T24')" onmouseout="rh()">PRE_TOPC:24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:7_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1_2"/><span class="lab"><font color="Green" title="E6">A8</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">BB</font> = <font color="Maroon" title="c2">BB</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:7_1_2"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">BB</font>
 
;<br/>
<a NAME="E4:7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A9</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font></span>)</span> <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D1" onmouseover="rs('tops_2/D1')" onmouseout="rh()">TOPS_2:def 1</a></span>;<br/>
<a NAME="E5:7_1_2"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
;<br/>
<a NAME="E6:7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A10</font></span>: 
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1_2"><span class="lab"><font color="Green" title="E7">A9</font></span></a>, <a class="ref" href="pre_topc.html#D3" onmouseover="rs('pre_topc/D3')" onmouseout="rh()">PRE_TOPC:def 3</a></span>;<br/>
<a NAME="E7:7_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">x</font>
 
;<br/>
<a NAME="E8:7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/>
<a NAME="E9:7_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:7_1_2"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_2"><span class="lab"><font color="Green" title="E6">A8</font></span></a>, <a class="txt" href="#E6:7_1_2"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="ref" href="pre_topc.html#T15" onmouseover="rs('pre_topc/T15')" onmouseout="rh()">PRE_TOPC:15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:7_1"/>
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N</font> being   <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b1">N</font> holds <br/> for <font color="Olive" title="b2">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> holds  <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b2">V</font></span><br/>

<span class="kw">thus </span><a NAME="E4:7_1"/>
 for <font color="Olive" title="b1">N</font> being   <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b1">N</font> holds <br/> for <font color="Olive" title="b2">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> holds  <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b2">V</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">N</font> be   <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c4">N</font> implies  for <font color="Olive" title="b1">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> holds  <font color="Maroon" title="c4">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><br/>

<span class="kw">assume </span><a NAME="E1:7_1_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c4">N</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> holds  <font color="Maroon" title="c4">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><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E6">A11</font></span>: <font color="Maroon" title="c3">x</font> = 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_2.html#D1" onmouseover="rs('waybel_2/D1')" onmouseout="rh()">WAYBEL_2:def 1</a></span>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> (<font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c1">T</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D5" onmouseover="rs('yellow_2/D5')" onmouseout="rh()">YELLOW_2:def 5</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c5">V</font></span><br/>

<a NAME="E3:7_1_3"/><span class="lab"><font color="Green" title="E7">A12</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" onmouseover="rs('connsp_2/D1')" onmouseout="rh()">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E4:7_1_3"/>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">BB</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#T23" onmouseover="rs('yellow_9/T23')" onmouseout="rh()">YELLOW_9:23</a></span>;<br/>
<a NAME="E5:7_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A13</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">BB</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#T22" onmouseover="rs('yellow_9/T22')" onmouseout="rh()">YELLOW_9:22</a></span>;<br/>
<a NAME="E6:7_1_3"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">Y</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E8:7_1_3"/><span class="lab"><font color="Green" title="E9">A14</font></span>: 
<font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">BB</font>
 <span class="kw">and </span><br/><a NAME="E9:7_1_3"/><span class="lab"><font color="Green" title="E10">A15</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_1_3"><span class="lab"><font color="Green" title="E8">A13</font></span></a>, <a class="ref" href="cantor_1.html#D1" onmouseover="rs('cantor_1/D1')" onmouseout="rh()">CANTOR_1:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">Y1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E11:7_1_3"/><span class="lab"><font color="Green" title="E11">A16</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Y1</font>
 <span class="kw">and </span><br/><a NAME="E12:7_1_3"/><span class="lab"><font color="Green" title="E12">A17</font></span>: 
<font color="Maroon" title="c7">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_3"><span class="lab"><font color="Green" title="E7">A12</font></span></a>, <a class="txt" href="#E9:7_1_3"><span class="lab"><font color="Green" title="E10">A15</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">Z</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E14:7_1_3"/><span class="lab"><font color="Green" title="E13">A18</font></span>: 
<font color="Maroon" title="c8">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">BB</font>
 <span class="kw">and </span><br/><a NAME="E15:7_1_3"/><span class="lab"><font color="Green" title="E14">A19</font></span>: 
<font color="Maroon" title="c8">Z</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">and </span><br/><a NAME="E16:7_1_3"/><span class="lab"><font color="Green" title="E15">A20</font></span>: 
<font color="Maroon" title="c7">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:7_1_3"><span class="lab"><font color="Green" title="E9">A14</font></span></a>, <a class="txt" href="#E12:7_1_3"><span class="lab"><font color="Green" title="E12">A17</font></span></a>, <a class="ref" href="cantor_1.html#D3" onmouseover="rs('cantor_1/D3')" onmouseout="rh()">CANTOR_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">rngN</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> (<font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E18:7_1_3"/>
<font color="Maroon" title="c9">rngN</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_2.html#T18" onmouseover="rs('waybel_2/T18')" onmouseout="rh()">WAYBEL_2:18</a></span>;<br/>
<a NAME="E19:7_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c9">rngN</font> &amp; (  for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c9">rngN</font> holds <br/><font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D39" onmouseover="rs('waybel_0/D39')" onmouseout="rh()">WAYBEL_0:def 39</a></span>;<br/>
<a NAME="E20:7_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A21</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c9">rngN</font>,<font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T15" onmouseover="rs('yellow_0/T15')" onmouseout="rh()">YELLOW_0:15</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">D1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; (  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b3">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">i</font> holds <br/><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">D1</font> ) );<br/>
<a NAME="E21:7_1_3"/><span class="lab"><font color="Green" title="E17">A22</font></span>: 
 for <font color="Olive" title="b1">Q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/> ex <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">b</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="c4">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Q</font>,<font color="Olive" title="b2">b</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">Q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> implies  ex <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">b</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="c4">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">Q</font>,<font color="Olive" title="b1">b</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1_3_2"/><span class="lab"><font color="Green" title="E18">A23</font></span>: 
<font color="Maroon" title="c10">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">b</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="c4">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">Q</font>,<font color="Olive" title="b1">b</font>] )</span><br/>

<a NAME="E2:7_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">BB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_1_3"><span class="lab"><font color="Green" title="E13">A18</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:7_1_3_2"/><span class="lab"><font color="Green" title="E19">A24</font></span>: 
<font color="Maroon" title="c10">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">v1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>
<a NAME="E5:7_1_3_2"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">v1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_1_3"><span class="lab"><font color="Green" title="E11">A16</font></span></a>, <a class="txt" href="#E16:7_1_3"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="txt" href="#E1:7_1_3_2"><span class="lab"><font color="Green" title="E18">A23</font></span></a>, <a class="txt" href="#E4:7_1_3_2"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="ref" href="setfam_1.html#T43" onmouseover="rs('setfam_1/T43')" onmouseout="rh()">SETFAM_1:43</a></span>;<br/>
<a NAME="E6:7_1_3_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E7:7_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A25</font></span>: 
 not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/>
<a NAME="E8:7_1_3_2"/>
 not <font color="Maroon" title="c9">rngN</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">v1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:7_1_3_2_1"/><span class="lab"><font color="Green" title="E21">A26</font></span>: 
<font color="Maroon" title="c9">rngN</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">v1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:7_1_3_2_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T34" onmouseover="rs('waybel_0/T34')" onmouseout="rh()">WAYBEL_0:34</a></span>;<br/>
<a NAME="E3:7_1_3_2_1"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c9">rngN</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">v1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:7_1_3"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="txt" href="#E1:7_1_3_2_1"><span class="lab"><font color="Green" title="E21">A26</font></span></a>, <a class="ref" href="yellow_0.html#T34" onmouseover="rs('yellow_0/T34')" onmouseout="rh()">YELLOW_0:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:7_1_3_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1_3"><span class="lab"><font color="Green" title="E6">A11</font></span></a>, <a class="txt" href="#E7:7_1_3_2"><span class="lab"><font color="Green" title="E20">A25</font></span></a>, <a class="ref" href="waybel_0.html#T34" onmouseover="rs('waybel_0/T34')" onmouseout="rh()">WAYBEL_0:34</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:7_1_3_2"/><span class="lab"><font color="Green" title="E21">A27</font></span>: 
<font color="Maroon" title="c12">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">rngN</font>
 <span class="kw">and </span><br/><a NAME="E11:7_1_3_2"/><span class="lab"><font color="Green" title="E22">A28</font></span>: 
 not <font color="Maroon" title="c12">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">v1</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">w9</font> = <font color="Maroon" title="c12">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:7_1_3_2"><span class="lab"><font color="Green" title="E21">A27</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E14:7_1_3_2"/><span class="lab"><font color="Green" title="E23">A29</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E15:7_1_3_2"/><span class="lab"><font color="Green" title="E24">A30</font></span>: 
<font color="Maroon" title="c13">w9</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="c4">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:7_1_3_2"><span class="lab"><font color="Green" title="E21">A27</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">i</font> = <font color="Maroon" title="c14">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_1_3_2"><span class="lab"><font color="Green" title="E23">A29</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><br/><a NAME="E18:7_1_3_2"/><span class="lab"><font color="Green" title="E25">A31</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>  st <font color="Maroon" title="c16">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> holds <br/><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c15">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T11" onmouseover="rs('waybel_0/T11')" onmouseout="rh()">WAYBEL_0:11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">b</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">b</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="c4">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">Q</font>,<font color="Maroon" title="c16">b</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E19:7_1_3_2"/>
<font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">Q</font>,<font color="Maroon" title="c16">b</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c17">QQ</font> = <font color="Maroon" title="c10">Q</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">QQ</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">QQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Q</font> &amp; (  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font> &amp; <font color="Olive" title="b2">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">QQ</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E21:7_1_3_2"/>
<font color="Maroon" title="c17">QQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font> &amp; <font color="Olive" title="b2">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">QQ</font></span><br/>

<span class="kw">thus </span><a NAME="E22:7_1_3_2"/>
 for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font> &amp; <font color="Olive" title="b2">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">QQ</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_3_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">j</font>, <font color="Maroon" title="c19">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font> &amp; <font color="Maroon" title="c19">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c18">j</font> implies <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">QQ</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7_1_3_2_2"/><span class="lab"><font color="Green" title="E26">A32</font></span>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font>
 <span class="kw">and </span><br/><a NAME="E2:7_1_3_2_2"/><span class="lab"><font color="Green" title="E27">A33</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c18">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">QQ</font></span><br/>

<a NAME="E3:7_1_3_2_2"/><span class="lab"><font color="Green" title="E28">A34</font></span>: 
<font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c15">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:7_1_3_2"><span class="lab"><font color="Green" title="E25">A31</font></span></a>, <a class="txt" href="#E1:7_1_3_2_2"><span class="lab"><font color="Green" title="E26">A32</font></span></a>, <a class="txt" href="#E2:7_1_3_2_2"><span class="lab"><font color="Green" title="E27">A33</font></span></a></span>;<br/>
<a NAME="E4:7_1_3_2_2"/>
 not <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c15">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_1_3_2"><span class="lab"><font color="Green" title="E22">A28</font></span></a>, <a class="txt" href="#E15:7_1_3_2"><span class="lab"><font color="Green" title="E24">A30</font></span></a>, <a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/>
<a NAME="E5:7_1_3_2_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_3_2_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="orders_2.html#T3" onmouseover="rs('orders_2/T3')" onmouseout="rh()">ORDERS_2:3</a></span>;<br/>
<a NAME="E6:7_1_3_2_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c11">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:7_1_3_2_2"/>
<font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">QQ</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1_3_2"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c10">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E23:7_1_3"/><span class="lab"><font color="Green" title="E18">A35</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Z</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">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="c4">N</font> )
 <span class="kw">and </span><br/><a NAME="E24:7_1_3"/><span class="lab"><font color="Green" title="E19">A36</font></span>: 
 for <font color="Olive" title="b1">Q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Q</font>,<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Q</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S6" onmouseover="rs('funct_1/S6')" onmouseout="rh()">FUNCT_1:sch 6</a>(<a class="txt" href="#E21:7_1_3"><span class="lab"><font color="Green" title="E17">A22</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">rngf</font> =  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">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="c4">N</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:7_1_3"><span class="lab"><font color="Green" title="E14">A19</font></span></a>, <a class="txt" href="#E23:7_1_3"><span class="lab"><font color="Green" title="E18">A35</font></span></a>, <a class="ref" href="finset_1.html#T8" onmouseover="rs('finset_1/T8')" onmouseout="rh()">FINSET_1:8</a></span>;<br/>
<a NAME="E26:7_1_3"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">N</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D6" onmouseover="rs('waybel_0/D6')" onmouseout="rh()">WAYBEL_0:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><br/><a NAME="E28:7_1_3"/>
<font color="Maroon" title="c12">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E29:7_1_3"/><span class="lab"><font color="Green" title="E20">A37</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c11">rngf</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T1" onmouseover="rs('waybel_0/T1')" onmouseout="rh()">WAYBEL_0:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">k</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D11" onmouseover="rs('waybel_0/D11')" onmouseout="rh()">WAYBEL_0:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font>  holds <br/>(  not <font color="Maroon" title="c12">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">V</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">k1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">k1</font> or <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E30:7_1_3"/><span class="lab"><font color="Green" title="E21">A38</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">k1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">V</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_3_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Q</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Q</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">Q</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">Z</font> implies <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">Q</font> )</span><br/><span class="kw">assume </span><a NAME="E1:7_1_3_3"/><span class="lab"><font color="Green" title="E22">A39</font></span>: 
<font color="Maroon" title="c14">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">Q</font></span><br/><a NAME="E2:7_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A40</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">rngf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:7_1_3"><span class="lab"><font color="Green" title="E18">A35</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">Q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> ;<br/><a NAME="E4:7_1_3_3"/><span class="lab"><font color="Green" title="E24">A41</font></span>: 
<font color="Maroon" title="c15">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:7_1_3"><span class="lab"><font color="Green" title="E20">A37</font></span></a>, <a class="txt" href="#E2:7_1_3_3"><span class="lab"><font color="Green" title="E23">A40</font></span></a></span>;<br/><a NAME="E5:7_1_3_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">Q</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:7_1_3"><span class="lab"><font color="Green" title="E19">A36</font></span></a>, <a class="txt" href="#E1:7_1_3_3"><span class="lab"><font color="Green" title="E22">A39</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:7_1_3_3"/>
<font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:7_1_3"><span class="lab"><font color="Green" title="E21">A38</font></span></a>, <a class="ref" href="orders_2.html#T3" onmouseover="rs('orders_2/T3')" onmouseout="rh()">ORDERS_2:3</a>, <a class="txt" href="#E4:7_1_3_3"><span class="lab"><font color="Green" title="E24">A41</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E32:7_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A42</font></span>: 
<font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:7_1_3"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="ref" href="setfam_1.html#T43" onmouseover="rs('setfam_1/T43')" onmouseout="rh()">SETFAM_1:43</a></span>;<br/>
<a NAME="E33:7_1_3"/>
<font color="Maroon" title="c7">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_1_3"><span class="lab"><font color="Green" title="E12">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/>
<a NAME="E34:7_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A43</font></span>: 
<font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_1_3"><span class="lab"><font color="Green" title="E10">A15</font></span></a>, <a class="txt" href="#E32:7_1_3"><span class="lab"><font color="Green" title="E22">A42</font></span></a></span>;<br/>
<a NAME="E35:7_1_3"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T16" onmouseover="rs('tops_1/T16')" onmouseout="rh()">TOPS_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:7_1_3"/>
<font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:7_1_3"><span class="lab"><font color="Green" title="E23">A43</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:7"/>
<font color="Maroon" title="c1">T</font> is  <a href="waybel32.html#V2" title="WAYBEL32:attr.2">order_consistent</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
