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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X0</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="tsp_2.html#NM1" title="TSP_2:NM.1">maximal_Kolmogorov_subspace</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">X0</font>  st <font color="Olive" title="b2">r</font> is  <a href="borsuk_1.html#V3" title="BORSUK_1:attr.3">being_a_retraction</a>  holds <br/>for <font color="Olive" title="b3">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">X0</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> holds <br/><font color="Olive" title="b2">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Olive" title="b3">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X0</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="tsp_2.html#NM1" title="TSP_2:NM.1">maximal_Kolmogorov_subspace</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font>  st <font color="Olive" title="b1">r</font> is  <a href="borsuk_1.html#V3" title="BORSUK_1:attr.3">being_a_retraction</a>  holds <br/>for <font color="Olive" title="b2">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">X0</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> holds <br/><font color="Olive" title="b1">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Olive" title="b2">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> is  <a href="borsuk_1.html#V3" title="BORSUK_1:attr.3">being_a_retraction</a>  implies for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">X0</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Olive" title="b1">a</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c3">r</font> is  <a href="borsuk_1.html#V3" title="BORSUK_1:attr.3">being_a_retraction</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">X0</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Olive" title="b1">a</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">X0</font>  st <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">X0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font> )</span><br/>



<span class="kw">assume </span><a NAME="E3:38"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font></span><br/>

<a NAME="E4:38"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_1"/>
not <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:38_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:38_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
not <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">s</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>;<br/>
<a NAME="E6:38_1"/>
<font color="Maroon" title="c3">r</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></span>;<br/>
<a NAME="E7:38_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E8:38_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T24" title="ZFMISC_1:th.24">ZFMISC_1:24</a></span>;<br/>
<a NAME="E9:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="tsp_2.html#E34"><span class="lab"><font color="Green" title="E24">Lm3</font></span></a></span>;<br/>
<a NAME="E10:38_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c8">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E11:38_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c8">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E12:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#T23" title="TEX_4:th.23">TEX_4:23</a></span>;<br/>
<a NAME="E13:38_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c8">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#T20" title="TEX_4:th.20">TEX_4:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:38_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E12:38_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:38"/>
 <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="tsp_2.html#E35"><span class="lab"><font color="Green" title="E25">Lm4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38"/>
<font color="Maroon" title="c3">r</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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