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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c1">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</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 <font color="Maroon" title="c2">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> #) &amp; <font color="Maroon" title="c1">S</font> is <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">sober</a> implies <font color="Maroon" title="c2">T</font> is <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">sober</a> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green" title="E11">A1</font></i>: 
 <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="c1">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</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 <font color="Maroon" title="c2">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> #)
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green" title="E12">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">irreducible</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b2">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Olive" title="b1">A</font> &amp; ( for <font color="Olive" title="b3">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b3">b</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D6" title="YELLOW_8:def.6">YELLOW_8:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">T</font> is <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">sober</a></span><br/>

<b>let </b><font color="Maroon" title="c3">B</font> be  <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">irreducible</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D6" title="YELLOW_8:def.6">YELLOW_8:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c3">B</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c3">B</font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">B</font> as  <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">irreducible</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="yellow14.html#T24" target="_self" title="YELLOW14:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E5:41"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E6:41"/><i><font color="Green" title="E14">A4</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">b</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font>
 <b>by </b><i><a class="txt" href="yellow14.html#E2:41"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c5">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> <b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c3">B</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<b>thus </b><a NAME="E8:41"/>
<font color="Maroon" title="c6">p</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow14.html#E5:41"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="yellow14.html#T25" target="_self" title="YELLOW14:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c7">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">q</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c7">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E10:41"/>
<font color="Maroon" title="c7">q</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font></span><br/>

<a NAME="E11:41"/><b>then </b>
<font color="Maroon" title="c8">b</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="yellow14.html#T25" target="_self" title="YELLOW14:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E12:41"/>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font>
 <b>by </b><i><a class="txt" href="yellow14.html#E6:41"><i><font color="Green" title="E14">A4</font></i></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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