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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">S</font> being <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds   <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">L</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> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ) iff for <font color="Olive" title="b1">S</font> being <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds   <a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">L</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>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">S</font> being <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds   <a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">L</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>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ) implies for <font color="Olive" title="b1">S</font> being <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds   <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">L</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> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span>
<div class="add"><b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E34">A1</font></i>: 
for <font color="Olive" title="b1">S</font> being <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds   <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">L</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> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds   <a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">L</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>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</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>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/><a NAME="E2:62_1"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="txt" href="#E1:62_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="waybel29.html#T35" target="_self" title="WAYBEL29:th.35">Th35</a></i>;<br/><a NAME="E3:62_1"/><b>then </b>
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="ref" href="waybel25.html#T13" title="WAYBEL25:th.13">WAYBEL25:13</a></i>;<br/><b>hence </b><a NAME="E4:62_1"/>
 <a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</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>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="ref" href="waybel25.html#T15" title="WAYBEL25:th.15">WAYBEL25:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:62"/><i><font color="Green" title="E34">A2</font></i>: 
for <font color="Olive" title="b1">S</font> being <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds   <a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">L</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>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds   <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">L</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> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Olive" title="b1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>

<b>let </b><font color="Maroon" title="c2">S</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</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> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>

<a NAME="E3:62"/>
 <a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</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>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<a NAME="E4:62"/><b>then </b>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="waybel25.html#D2" title="WAYBEL25:def.2">WAYBEL25:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:62"/>
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">L</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> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_9.html#T51" title="YELLOW_9:th.51">YELLOW_9:51</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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