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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel25.html#V1" target="_self">monotone-convergence</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/>



<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E1:60"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel25.html#D4" target="_self">WAYBEL25:def 4</a><br/>

<a NAME="E2:60"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</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>by </b><i><a class="txt" href="waybel25.html#E15"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<a NAME="E3:60"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</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>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E15"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM7">Function</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>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E2:60"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E5:60"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:60"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel25.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E6:60"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="ref" href="yellow14.html#T11">YELLOW14:11</a></i>;<br/>
<a NAME="E7:60"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E5:60"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E8:60"/><i><font color="Green">E39</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:60"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E9:60"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E7:60"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E10:60"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E6:60"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<a NAME="E11:60"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>
<a NAME="E12:60"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</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#E8:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel25.html#E9:60"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tops_2.html#T68">TOPS_2:68</a></i>;<br/>
<a NAME="E13:60"/><b>then </b>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font>, <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel25.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E14:60"/><b>then </b>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>, <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E5:60"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow14.html#T17">YELLOW14:17</a></i>;<br/>
<b>hence </b><a NAME="E15:60"/><i><font color="Green">E42</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>4</sub></font>, <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E7:60"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E16:60"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:60"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topgrp_1.html#T26">TOPGRP_1:26</a></i>;<br/>
<a NAME="E17:60"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="waybel_0.html#V18">sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E6:60"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<a NAME="E18:60"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>
<b>assume </b><a NAME="E19:60"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E20:60"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E21:60"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E8:60"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tops_2.html#T68">TOPS_2:68</a></i>;<br/>
<a NAME="E22:60"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E8:60"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E23:60"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E8:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel25.html#E9:60"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E24:60"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E15:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel25.html#E18:60"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E25:60"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E8:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel25.html#E9:60"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tops_2.html#T68">TOPS_2:68</a></i>;<br/>
<a NAME="E26:60"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E12:60"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel25.html#E16:60"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel25.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:60"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</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="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#E28:60"><i><font color="Green">E45</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/><b>thus </b><a NAME="E1:60_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="waybel25.html#E28:60"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E31:60"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div>
