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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">S</font> being <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> holds   <a href="waybel11.html#K5">sigma</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Olive">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Olive">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/><a NAME="E2:61_1"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon">L</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">SL</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">SS</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>;<br/>





<a NAME="E1:61_1_1"/>
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SS</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">SS</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">S</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">S</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">SL</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E2:61_1_1"/><b>then </b>
(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">S</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">S</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SS</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SS</font> #) &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font> #) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:61_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:61_1_1"/>
 <a href="waybel11.html#K5">sigma</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:61_1"/>
for <font color="Olive">S</font> being <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> holds   <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Olive">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Olive">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Olive">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">S</font> being <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> holds   <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Olive">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Olive">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Olive">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<a NAME="E3:61"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:61"/>
 <a href="waybel11.html#K5">sigma</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <font color="Maroon">L</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>


</div>
