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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:80"/><i><font color="Green">E49</font></i>: 
 <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> #) <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>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #)
 ;<br/>

<a NAME="E2:80"/><i><font color="Green">E50</font></i>: 
 <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21">YELLOW_6:def 21</a></i>;<br/>
<a NAME="E3:80"/><i><font color="Green">E51</font></i>: 
 <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21">YELLOW_6:def 21</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:80_1"/><i><font color="Green">E52</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>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:80_1"/><i><font color="Green">E53</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>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="waybel11.html#E2:80"><i><font color="Green">E50</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E4:80_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E5:80_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E7:80_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:80_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="waybel11.html#E3:80_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="yellow_6.html#D22">YELLOW_6:def 22</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel11.html#E1:80"><i><font color="Green">E49</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel11.html#E1:80"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E10:80_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="waybel11.html#E1:80"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel11.html#E4:80_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="waybel11.html#E7:80_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="waybel11.html#E54"><i><font color="Green">Lemma45</font></i></a>, <a class="txt" href="waybel11.html#E56"><i><font color="Green">Lemma47</font></i></a></i>;<br/><b>hence </b><a NAME="E11:80_1"/>
<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>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E3:80_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="yellow_6.html#D22">YELLOW_6:def 22</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:80"/><i><font color="Green">E52</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>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:80"/><i><font color="Green">E53</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>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="waybel11.html#E3:80"><i><font color="Green">E51</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E8:80"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E9:80"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E11:80"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E5:80"><i><font color="Green">E52</font></i></a>, <a class="txt" href="waybel11.html#E7:80"><i><font color="Green">E53</font></i></a>, <a class="ref" href="yellow_6.html#D22">YELLOW_6:def 22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel11.html#E1:80"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel11.html#E1:80"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E14:80"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:80"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel11.html#E8:80"><i><font color="Green">E54</font></i></a>, <a class="txt" href="waybel11.html#E11:80"><i><font color="Green">E55</font></i></a>, <a class="txt" href="waybel11.html#E54"><i><font color="Green">Lemma45</font></i></a>, <a class="txt" href="waybel11.html#E56"><i><font color="Green">Lemma47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:80"/>
<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>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E7:80"><i><font color="Green">E53</font></i></a>, <a class="ref" href="yellow_6.html#D22">YELLOW_6:def 22</a></i>;<br/>


</div>
