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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  implies  for <font color="Olive" title="b1">Y</font> being   <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a>  <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</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">assume </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being   <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a>  <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</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/>

<a NAME="E2:46"/>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>, <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>) <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T6" target="_self" onmouseover="rs('waybel26/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">XS</font> =  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="waybel15.html#T9" onmouseover="rs('waybel15/T9')" onmouseout="rh()">WAYBEL15:9</a>, <a class="ref" href="waybel20.html#T18" onmouseover="rs('waybel20/T18')" onmouseout="rh()">WAYBEL20:18</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">Y</font> be   <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a>  <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c3">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="c3">Y</font>) is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E5:46"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c3">Y</font> <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Maroon" title="c4">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel18.html#T19" onmouseover="rs('waybel18/T19')" onmouseout="rh()">WAYBEL18:19</a></span>;<br/>
<a NAME="E6:46"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">M</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span> <a href="waybel18.html#K4" title="WAYBEL18:func.4">.</a> <font color="Olive" title="b1">i</font> is  <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">MS</font> = <font color="Maroon" title="c4">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span> as   <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a>  <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel18.html#T7" onmouseover="rs('waybel18/T7')" onmouseout="rh()">WAYBEL18:7</a></span>;<br/>
<a NAME="E8:46"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">M</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">XS</font></span>)</span> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Olive" title="b1">i</font> is   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<a NAME="E9:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A3</font></span>: 
( <font color="Maroon" title="c4">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">XS</font></span>)</span> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp; <font color="Maroon" title="c4">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">XS</font></span>)</span> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel20.html#T33" onmouseover="rs('waybel20/T33')" onmouseout="rh()">WAYBEL20:33</a></span>;<br/>
<a NAME="E10:46"/>
<font color="Maroon" title="c4">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span>, <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>) <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T35" target="_self" onmouseover="rs('waybel26/T35')" onmouseout="rh()">Th35</a>, <a class="ref" href="waybel_1.html#T6" onmouseover="rs('waybel_1/T6')" onmouseout="rh()">WAYBEL_1:6</a></span>;<br/>
<a NAME="E11:46"/><span class="kw">then </span>
(  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">MS</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="c5">MS</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="#E9:46"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="waybel15.html#T9" onmouseover="rs('waybel15/T9')" onmouseout="rh()">WAYBEL15:9</a>, <a class="ref" href="waybel20.html#T18" onmouseover="rs('waybel20/T18')" onmouseout="rh()">WAYBEL20:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:46"/>
(  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">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="c3">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="#E5:46"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="waybel26.html#T23" target="_self" onmouseover="rs('waybel26/T23')" onmouseout="rh()">Th23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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