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<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D6">ORDERS_2:def 6</a>,<a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_1_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:28_1_1"/><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:28_1_1"/><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E1:28_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E5:28_1_1"/>
 <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="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#D2" target="_self">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel25.html#E1:28_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E7:28_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:28_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:28_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E10:28_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="waybel25.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E11:28_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E3:28_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:28_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E14:28_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="waybel25.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E15:28_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> or ( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="waybel25.html#E9:28_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel25.html#E10:28_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel25.html#E13:28_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel25.html#E14:28_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="yellow14.html#T22">YELLOW14:22</a></i>;<br/>
<b>hence </b><a NAME="E16:28_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="t_0topsp.html#D7">T_0TOPSP:def 7</a></i>;<br/>


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