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<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">Y</font>, <font color="Olive" title="b2">Z</font> being  <a href="waybel25.html#V1" title="WAYBEL25:attr.1">monotone-convergence</a>  <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a>  st <font color="Olive" title="b1">Y</font> <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Olive" title="b2">Z</font> &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">Z</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">Z</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  holds <br/>(  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">Y</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">Y</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be   <a href="waybel25.html#V1" title="WAYBEL25:attr.1">monotone-convergence</a>  <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a>; <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="c2">Y</font> <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Maroon" title="c3">Z</font> &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  implies (  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:33"/>
<font color="Maroon" title="c2">Y</font> <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Maroon" title="c3">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  or  not  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  or (  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">S</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">Z</font><span class="kw"> such that </span><br/><a NAME="E3:33"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c4">S</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a> <font color="Maroon" title="c3">Z</font>
 <span class="kw">and </span><br/><a NAME="E4:33"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c2">Y</font> <a href="t_0topsp.html#R1" title="T_0TOPSP:pred.1">are_homeomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow16.html#T59" title="YELLOW16:th.59">YELLOW16:59</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:33"/>
(  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )</span><br/>

<a NAME="E6:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A3</font></span>: 
(  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">S</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">S</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="waybel26.html#T23" target="_self" title="WAYBEL26:th.23">Th23</a></span>;<br/>
<a NAME="E7:33"/>
 <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">S</font>, <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="waybel26.html#T22" target="_self" title="WAYBEL26:th.22">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:33"/>
(  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="waybel15.html#T11" title="WAYBEL15:th.11">WAYBEL15:11</a>, <a class="ref" href="waybel20.html#T18" title="WAYBEL20:th.18">WAYBEL20:18</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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