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

<span class="kw">let </span><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">N</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">LM</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">LN</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">N</font>  st  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">N</font></span>)</span> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  holds <br/>the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Olive" title="b1">LM</font>,<font color="Olive" title="b2">LN</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>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">LM</font> be   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">M</font>; <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">LN</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">N</font>  st  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">N</font></span>)</span> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  holds <br/>the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c3">LM</font>,<font color="Olive" title="b1">LN</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>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">LN</font> be   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">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_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">N</font></span>)</span> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  implies the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c3">LM</font>,<font color="Maroon" title="c4">LN</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>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">N</font></span>)</span> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c3">LM</font>,<font color="Maroon" title="c4">LN</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>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">SMN</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</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">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">lMN</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</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">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">LMN</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</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">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>
<a NAME="E5:33"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c3">LM</font>,<font color="Maroon" title="c4">LN</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>  <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 <font color="Maroon" title="c7">LMN</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">LMN</font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c8">lN</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">N</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">lM</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">M</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">SN</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">N</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">SM</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E5:33_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c4">LN</font> is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c10">SN</font>,<font color="Maroon" title="c8">lN</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#T29" title="WAYBEL19:th.29">WAYBEL19:29</a></span>;<br/>
<a NAME="E6:33_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">lN</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c8">lN</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">N</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E7:33_1"/>
<span class="p1">(<span class="default"><a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <font color="Maroon" title="c7">LMN</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c7">LMN</font></span>)</span> is   <a href="cantor_1.html#NM2" title="CANTOR_1:NM.2">prebasis</a> of <font color="Maroon" title="c7">LMN</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#D3" title="WAYBEL19:def.3">WAYBEL19:def 3</a></span>;<br/>
<a NAME="E8:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<span class="p1">(<span class="default"><a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <font color="Maroon" title="c7">LMN</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c7">LMN</font></span>)</span> is   <a href="cantor_1.html#NM2" title="CANTOR_1:NM.2">prebasis</a> of <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 <font color="Maroon" title="c7">LMN</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">LMN</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow12.html#T33" title="YELLOW12:th.33">YELLOW12:33</a></span>;<br/>
<a NAME="E9:33_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">LM</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">LM</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">M</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E10:33_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">LN</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">LN</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">N</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E11:33_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">LMN</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">LMN</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</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">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</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">M</font>,<font color="Maroon" title="c2">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="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E25">A9</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="c3">LM</font>,<font color="Maroon" title="c4">LN</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> = 
<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="c3">LM</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">LN</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/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">LMN</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E10:33_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E11:33_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E26">A10</font></span>: the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c9">lM</font>,<font color="Maroon" title="c8">lN</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> = 
 <a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">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="waybel19.html#T15" title="WAYBEL19:th.15">WAYBEL19:15</a></span>
<br/>.= 
 <a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <font color="Maroon" title="c7">LMN</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="waybel19.html#T3" title="WAYBEL19:th.3">WAYBEL19:3</a></span>
;<br/>
<a NAME="E14:33_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">SN</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">SN</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">N</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">SM</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">SM</font> #) = 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">M</font> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>
<br/>.= 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</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> <font color="Maroon" title="c1">M</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c1">M</font></span>)</span> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>
;<br/>
<a NAME="E16:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A12</font></span>: 
 <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 <font color="Maroon" title="c11">SM</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c11">SM</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> <font color="Maroon" title="c1">M</font></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> <font color="Maroon" title="c1">M</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel29.html#T17" title="WAYBEL29:th.17">WAYBEL29:17</a></span>;<br/>
<a NAME="E17:33_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">lM</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c9">lM</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">M</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">SN</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">SN</font> #) = 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">N</font> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>
<br/>.= 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</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> <font color="Maroon" title="c2">N</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">N</font></span>)</span> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>
;<br/>
<a NAME="E19:33_1"/><span class="kw">then </span>
 <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 <font color="Maroon" title="c10">SN</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c10">SN</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> <font color="Maroon" title="c2">N</font></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> <font color="Maroon" title="c2">N</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel29.html#T17" title="WAYBEL29:th.17">WAYBEL29:17</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E30">A14</font></span>: the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c11">SM</font>,<font color="Maroon" title="c10">SN</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> = 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</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="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" title="WAYBEL29:func.1">Sigma</a> <font color="Maroon" title="c2">N</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:33_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="waybel29.html#T10" title="WAYBEL29:th.10">WAYBEL29:10</a></span>
<br/>.= 
 <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="c1">M</font>,<font color="Maroon" title="c2">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="txt" href="#E1:33"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="waybel29.html#T34" title="WAYBEL29:th.34">WAYBEL29:34</a></span>
<br/>.= 
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c7">LMN</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="yellow_9.html#T52" title="YELLOW_9:th.52">YELLOW_9:52</a></span>
;<br/>
<a NAME="E21:33_1"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">SM</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">SM</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">M</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E22:33_1"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
<font color="Maroon" title="c3">LM</font> is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c11">SM</font>,<font color="Maroon" title="c9">lM</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#T29" title="WAYBEL19:th.29">WAYBEL19:29</a></span>;<br/>
<a NAME="E23:33_1"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c3">LM</font>,<font color="Maroon" title="c4">LN</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c11">SM</font>,<font color="Maroon" title="c10">SN</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c9">lM</font>,<font color="Maroon" title="c8">lN</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="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E21:33_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E14:33_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E17:33_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="yellow12.html#T50" title="YELLOW12:th.50">YELLOW12:50</a></span>;<br/>
<a NAME="E24:33_1"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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 <font color="Maroon" title="c7">LMN</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">LMN</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="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c11">SM</font>,<font color="Maroon" title="c10">SN</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> 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="c9">lM</font>,<font color="Maroon" title="c8">lN</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="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="yellow_9.html#D6" title="YELLOW_9:def.6">YELLOW_9:def 6</a></span>;<br/>
<a NAME="E25:33_1"/><span class="kw">then </span>
 <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 <font color="Maroon" title="c7">LMN</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">LMN</font> #) is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c11">SM</font>,<font color="Maroon" title="c10">SN</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c9">lM</font>,<font color="Maroon" title="c8">lN</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="txt" href="#E8:33_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="yellow_9.html#D6" title="YELLOW_9:def.6">YELLOW_9:def 6</a>, <a class="txt" href="#E20:33_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E13:33_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:33_1"/>
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c3">LM</font>,<font color="Maroon" title="c4">LN</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>  <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 <font color="Maroon" title="c7">LMN</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">LMN</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:33_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E21:33_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E14:33_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E17:33_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="yellow12.html#T49" title="YELLOW12:th.49">YELLOW12:49</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="E6:33"/>
<font color="Maroon" title="c7">LMN</font> is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c5">SMN</font>,<font color="Maroon" title="c6">lMN</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#T29" title="WAYBEL19:th.29">WAYBEL19:29</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">R</font> = <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c3">LM</font>,<font color="Maroon" title="c4">LN</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> as    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c5">SMN</font>,<font color="Maroon" title="c6">lMN</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="yellow12.html#T47" title="YELLOW12:th.47">YELLOW12:47</a></span>;<br/>
<a NAME="E8:33"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c3">LM</font>,<font color="Maroon" title="c4">LN</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> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c8">R</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E9:33"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c3">LM</font>,<font color="Maroon" title="c4">LN</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>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">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="waybel19.html#T34" title="WAYBEL19:th.34">WAYBEL19:34</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>
