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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><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>;<br/>



<b>assume </b><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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">S</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">Z</font><b> such that </b><br/><a NAME="E3:33"/><i><font color="Green" title="E27">A1</font></i>: 
( <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> &amp; <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>  )
 <b>by </b><i><a class="ref" href="yellow16.html#T59" title="YELLOW16:th.59">YELLOW16:59</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green" title="E28">A2</font></i>: 
 <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> 
 
<b>by </b><i><a class="txt" href="waybel26.html#E3:33"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="waybel26.html#T22" target="_self" title="WAYBEL26:th.22">Th22</a></i>;<br/>
<b>assume </b><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> )
 ;<br/>

<a NAME="E6:33"/><b>then </b>
(  <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> )
 
<b>by </b><i><a class="txt" href="waybel26.html#E3:33"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="waybel26.html#T23" target="_self" title="WAYBEL26:th.23">Th23</a></i>;<br/>
<b>hence </b><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="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> )
 <b>by </b><i><a class="txt" href="waybel26.html#E4:33"><i><font color="Green" title="E28">A2</font></i></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></i>;<br/>


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