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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:48"/><i><font color="Green">E40</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel18.html#V2">injective</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#R1">is_Retract_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#T20">WAYBEL18:20</a></i>;<br/>
<a NAME="E4:48"/>
 <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel26.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E5:48"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>, <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T7">WAYBEL_1:7</a></i>;<br/>
<a NAME="E6:48"/>
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a>  <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel26.html#T6" target="_self">Th6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a>  as  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="orders_2.html#NM1">Poset</a> <b>by </b><i><a class="txt" href="waybel26.html#E1:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel15.html#T11">WAYBEL15:11</a>, <a class="ref" href="waybel20.html#T18">WAYBEL20:18</a></i>;<br/>
<a NAME="E8:48"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <a href="waybel18.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel18.html#V2">injective</a>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> as  <a href="waybel18.html#V2">injective</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a> <b>by </b><i><a class="ref" href="waybel18.html#T8">WAYBEL18:8</a></i>;<br/>
<a NAME="E10:48"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="yellow16.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E11:48"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="waybel_3.html#V3">continuous</a> )
 
<b>by </b><i><a class="ref" href="waybel20.html#T34">WAYBEL20:34</a></i>;<br/>
<a NAME="E12:48"/><b>then </b>
(  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> is <a href="lattice3.html#V3">complete</a> &amp;  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_3.html#V3">continuous</a> )
 
<b>by </b><i><a class="txt" href="waybel26.html#E5:48"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel15.html#T11">WAYBEL15:11</a>, <a class="ref" href="waybel20.html#T18">WAYBEL20:18</a></i>;<br/>
<b>hence </b><a NAME="E13:48"/>
(  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V3">complete</a> &amp;  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_3.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="waybel26.html#E3:48"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel26.html#T24" target="_self">Th24</a></i>;<br/>


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