<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="pre_topc.html#V5">continuous</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>;<br/>

<a NAME="E1:48_1_1"/>
 <a href="relset_1.html#K4">dom</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>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:48_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>thus </b><a NAME="E3:48_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="tsp_2.html#K3" target="_self">Stone-retraction</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies 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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="tsp_2.html#K3" target="_self">Stone-retraction</a> <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>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>assume </b><a NAME="E3:48_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:48_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="tsp_2.html#V1" target="_self">maximal_T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_2.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:48_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="tsp_1.html#V1">T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_2.html#D4" target="_self">Def4</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E6:48_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1_1_1_1"/>
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> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:48_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:48_1_1_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E5:48_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tsp_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:48_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:48_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tex_4.html#D11">TEX_4:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:48_1_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E9:48_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</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="E11:48_1_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E12:48_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>10</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="tsp_2.html#E9:48_1_1_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E13:48_1_1_1_1_1"/>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E8:48_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="tsp_2.html#E11:48_1_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tex_4.html#T23">TEX_4:23</a></i>;<br/>
<a NAME="E14:48_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="tsp_2.html#E3:48_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="tsp_2.html#E4:48_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tsp_2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E15:48_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/>
<b>hence </b><a NAME="E16:48_1_1_1_1_1"/>
<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#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E12:48_1_1_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:48_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:48_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:48_1_1_1_2"/>
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> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1_2_1"/><i><font color="Green">E39</font></i>: 
<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#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:48_1_1_1_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:48_1_1_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E6:48_1_1_1_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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="tsp_2.html#E4:48_1_1_1_2_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E8:48_1_1_1_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="tsp_2.html#E3:48_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="tsp_2.html#E6:48_1_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tsp_2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E9:48_1_1_1_2_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E5:48_1_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:48_1_1_1_2_1"/><b>then </b>
 <a href="tex_4.html#K3">MaxADSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T33">TEX_4:33</a></i>;<br/>
<a NAME="E11:48_1_1_1_2_1"/><b>then </b>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T30">TEX_4:30</a></i>;<br/>
<a NAME="E12:48_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<b>hence </b><a NAME="E13:48_1_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tsp_2.html#E8:48_1_1_1_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:48_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:48_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E6:48_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:48_1_1"/><i><font color="Green">E35</font></i>: 
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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<a NAME="E5:48_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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:48_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</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="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:48_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <span class="p2"><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></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</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>by </b><i><a class="txt" href="tsp_2.html#E4:48_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="tsp_2.html#E1:48_1_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E3:48_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</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>by </b><i><a class="ref" href="tex_4.html#T30">TEX_4:30</a></i>;<br/>
<b>hence </b><a NAME="E4:48_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tsp_2.html#E2:48_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:48_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="tsp_2.html#K3" target="_self">Stone-retraction</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tsp_2.html#D10" target="_self">Def10</a></i>;<br/>


</div>
