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

<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>   <a href="yellow13.html#V2" title="YELLOW13:attr.2">topological_semilattice</a>  <a href="waybel19.html#NM1" title="WAYBEL19:NM.1">TopPoset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>  st  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">C</font> is  <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>  holds <br/> <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">C</font></span>)</span> is  <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">C</font> is  <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>  implies  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span> is  <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:53"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">C</font> is  <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span> is  <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">A</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D16" title="YELLOW_0:def.16">YELLOW_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> or  not <font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> or  not  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">N</font> or  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">N</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_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:53"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:53"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">N</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_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span><br/>

<a NAME="E5:53"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E6:53"/>
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="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>  holds <font color="Olive" title="b1">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c7">NN</font> = <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font></span><br/>

<a NAME="E1:53_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">xy</font> = <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></span>;<br/>
<a NAME="E3:53_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> =  <a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">N</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<font color="Maroon" title="c1">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E34">A6</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="c9">xy</font> = 
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>

<br/>.= 
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_2.html#D4" title="WAYBEL_2:def.4">WAYBEL_2:def 4</a></span>
;<br/>
<a NAME="E6:53_1"/>
<font color="Maroon" title="c10">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow13.html#D5" title="YELLOW13:def.5">YELLOW13:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">H</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c9">xy</font><span class="kw"> such that </span><br/><a NAME="E8:53_1"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="borsuk_1.html#D3" title="BORSUK_1:def.3">BORSUK_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">A</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E10:53_1"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">H</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="c12">A</font>
 <span class="kw">and </span><br/><a NAME="E11:53_1"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> holds <br/> ex <font color="Olive" title="b2">X1</font>, <font color="Olive" title="b3">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> &amp; <font color="Olive" title="b2">X1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b3">Y1</font> 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="borsuk_1.html#T45" title="BORSUK_1:th.45">BORSUK_1:45</a></span>;<br/>
<a NAME="E12:53_1"/>
<font color="Maroon" title="c9">xy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:53_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">K</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:53_1"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c9">xy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">K</font>
 <span class="kw">and </span><br/><a NAME="E15:53_1"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<font color="Maroon" title="c13">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">Ix</font>, <font color="Maroon" title="c15">Iy</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><span class="kw"> such that </span><br/><a NAME="E17:53_1"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
<font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c14">Ix</font>,<font color="Maroon" title="c15">Iy</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E18:53_1"/><span class="lab"><font color="Green" title="E41">A13</font></span>: 
<font color="Maroon" title="c14">Ix</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E19:53_1"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
<font color="Maroon" title="c15">Iy</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E15:53_1"><span class="lab"><font color="Green" title="E39">A11</font></span></a></span>;<br/>
<a NAME="E20:53_1"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Ix</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:53_1"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="txt" href="#E17:53_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E21:53_1"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Iy</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:53_1"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="txt" href="#E17:53_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E22:53_1"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
<font color="Maroon" title="c14">Ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c14">Ix</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:53_1"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></span>;<br/>
<a NAME="E23:53_1"/>
<font color="Maroon" title="c15">Iy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">Iy</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:53_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">Iy</font> = <font color="Maroon" title="c15">Iy</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:53_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E25:53_1"/>
<font color="Maroon" title="c16">Iy</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="connsp_2.html#T34" title="CONNSP_2:th.34">CONNSP_2:34</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">iy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E27:53_1"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
<font color="Maroon" title="c17">iy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Iy</font>
 <span class="kw">and </span><br/><a NAME="E28:53_1"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
<font color="Maroon" title="c17">iy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">Ix</font> = <font color="Maroon" title="c14">Ix</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">x</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:53_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E22:53_1"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E30:53_1"/>
<font color="Maroon" title="c18">Ix</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="connsp_2.html#T34" title="CONNSP_2:th.34">CONNSP_2:34</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">ix</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E32:53_1"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
<font color="Maroon" title="c19">ix</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Ix</font>
 <span class="kw">and </span><br/><a NAME="E33:53_1"/><span class="lab"><font color="Green" title="E49">A21</font></span>: 
<font color="Maroon" title="c19">ix</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">ix</font> = <font color="Maroon" title="c19">ix</font>, <font color="Maroon" title="c21">iy</font> = <font color="Maroon" title="c17">iy</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:53_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E27:53_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:53_1_2"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c20">ix</font>,<font color="Maroon" title="c21">iy</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:53_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E32:53_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E27:53_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E2:53_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A22</font></span>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c20">ix</font>,<font color="Maroon" title="c21">iy</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:53_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E15:53_1"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">a</font> = <font color="Maroon" title="c20">ix</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">iy</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )</span><br/><a NAME="E3:53_1_2"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E4:53_1_2"/><span class="lab"><font color="Green" title="E52">A24</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/><a NAME="E5:53_1_2"/>
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">ix</font>,<font color="Maroon" title="c21">iy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">ix</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">iy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_2.html#D4" title="WAYBEL_2:def.4">WAYBEL_2:def 4</a></span>;<br/><a NAME="E6:53_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">ix</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">iy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_1_2"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="txt" href="#E3:53_1_2"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="txt" href="#E2:53_1_2"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E7:53_1_2"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:53_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font></span><br/><a NAME="E8:53_1_2"/><span class="lab"><font color="Green" title="E53">A25</font></span>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">ix</font>,<font color="Maroon" title="c21">iy</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T21" title="YELLOW_0:th.21">YELLOW_0:21</a></span>;<br/><a NAME="E9:53_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><a NAME="E10:53_1_2"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">ix</font>,<font color="Maroon" title="c21">iy</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:53_1_2"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E33:53_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="txt" href="#E28:53_1"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="yellow_0.html#D16" title="YELLOW_0:def.16">YELLOW_0:def 16</a></span>;<br/><span class="kw">hence </span><a NAME="E11:53_1_2"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E36:53_1"/>
<font color="Maroon" title="c8">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:53"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</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> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#T34" title="CONNSP_2:th.34">CONNSP_2:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:53"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">N</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_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
