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<div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">X0</font>;<br/>

<b>thus </b><a NAME="E1:44_1_1"/>
( <font color="Maroon">r</font> <a href="funct_2.html#R2">=</a>  <a href="tsp_2.html#K1" target="_self">Stone-retraction</a> <font color="Maroon">X</font>,<font color="Maroon">X0</font> implies for <font color="Olive">M</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> holds <br/>for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Olive">M</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">a</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">r</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">r</font> <a href="funct_2.html#R2">=</a>  <a href="tsp_2.html#K1" target="_self">Stone-retraction</a> <font color="Maroon">X</font>,<font color="Maroon">X0</font>
 ;<br/>

<b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:44_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/>

<a NAME="E3:44_1_1_1"/>
 <a href="tsp_2.html#K1" target="_self">Stone-retraction</a> <font color="Maroon">X</font>,<font color="Maroon">X0</font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:44_1_1_1"/>
for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">M</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">a</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">r</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , </i>;<br/>


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

<b>reconsider </b><font color="Maroon">M</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E3:44_1_1"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> holds <br/>for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Olive">A</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">a</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">r</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:44_1_1"/><b>then </b>
for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">M</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">a</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">r</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
;<br/>
<a NAME="E5:44_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:44_1_1"/>
<font color="Maroon">r</font> <a href="funct_2.html#R2">=</a>  <a href="tsp_2.html#K1" target="_self">Stone-retraction</a> <font color="Maroon">X</font>,<font color="Maroon">X0</font>
 <b>by </b><i/>;<br/>


</div>
