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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <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>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<a NAME="E3:8"/><i><font color="Green">E6</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</sub></font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E4:8"/><i><font color="Green">E7</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E5:8"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:8"><i><font color="Green">E5</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<i><font color="Green">E9</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="waybel25.html#E2:8"><i><font color="Green">E5</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>

;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:8"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> 
 ;<br/>
<a NAME="E9:8"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="txt" href="waybel25.html#E5:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel25.html#E6:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel25.html#E8:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E10:8"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:8"><i><font color="Green">E5</font></i></a>, <a class="ref" href="waybel_0.html#T68">WAYBEL_0:68</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="waybel_0.html#V18">sups-preserving</a> <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>by </b><i><a class="txt" href="waybel25.html#E2:8"><i><font color="Green">E5</font></i></a>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  as  <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel25.html#E8:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel25.html#E10:8"><i><font color="Green">E11</font></i></a>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<a NAME="E13:8"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
;<br/>
<a NAME="E14:8"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel25.html#E3:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="waybel25.html#E4:8"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E16:8"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:8"><i><font color="Green">E4</font></i></a>, <a class="txt" href="waybel25.html#E3:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="waybel25.html#E4:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel25.html#E13:8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel21.html#T6">WAYBEL21:6</a></i>;<br/>
<a NAME="E17:8"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T22">WAYBEL17:22</a></i>;<br/>
<a NAME="E18:8"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E3:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="waybel25.html#E4:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel25.html#E14:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel21.html#T6">WAYBEL21:6</a></i>;<br/>
<a NAME="E19:8"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T22">WAYBEL17:22</a></i>;<br/>
<i><font color="Green">E16</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>
<i><font color="Green">E17</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="waybel25.html#E1:8"><i><font color="Green">E4</font></i></a>, <a class="txt" href="waybel25.html#E2:8"><i><font color="Green">E5</font></i></a>, <a class="txt" href="waybel25.html#E4:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>4</sub></font>

;<br/>
<a NAME="E22:8"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:8"><i><font color="Green">E4</font></i></a>, <a class="txt" href="waybel25.html#E2:8"><i><font color="Green">E5</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<i><font color="Green">E19</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E24:8"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="waybel25.html#E5:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel25.html#E6:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="waybel25.html#E1:8"><i><font color="Green">E4</font></i></a>, <a class="txt" href="waybel25.html#E5:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel25.html#E21:8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E25:8"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E23:8"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E26:8"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="waybel25.html#E17:8"><i><font color="Green">E14</font></i></a>, <a class="txt" href="waybel25.html#E19:8"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel25.html#E20:8"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel25.html#E21:8"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel25.html#E22:8"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>


</div>
