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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c1">Y</font> is <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a> implies  <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Maroon" title="c1">Y</font> )</span><br/>

<b>given </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><a NAME="E2:33"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <b>and </b><br/><a NAME="E3:33"/><i><font color="Green" title="E28">A2</font></i>: 
for <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="urysohn1.html#D9" title="URYSOHN1:def.9">URYSOHN1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Maroon" title="c1">Y</font></span><br/>

<a NAME="E4:33"/>
(  ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> ) or  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> ) )
 
<b>by </b><i><a class="txt" href="#E2:33"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="tsp_1.html#D3" title="TSP_1:def.3">TSP_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E6:33"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E7:33"/><i><font color="Green" title="E30">A4</font></i>: 
( ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> ) or ( not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> ) )
 ;<br/>
<a NAME="E8:33"/><i><font color="Green" title="E31">A5</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">Y</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">Y</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Y</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">Y</font> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#D2" title="WAYBEL25:def.2">WAYBEL25:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">Y</font></span>)</span><b> such that </b><br/><a NAME="E10:33"/><i><font color="Green" title="E32">A6</font></i>: 
( ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> ) or ( not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> ) )
 <b>by </b><i><a class="txt" href="#E7:33"><i><font color="Green" title="E30">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">V</font> = <font color="Maroon" title="c4">V</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">Y</font></span>)</span> <b>by </b><i><a class="txt" href="#E6:33"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="#E8:33"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="tops_3.html#T76" title="TOPS_3:th.76">TOPS_3:76</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">c</font> =  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c7">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Y</font> as  <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">Y</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <b>by </b><i><a class="txt" href="#E6:33"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="yellow16.html#T48" title="YELLOW16:th.48">YELLOW16:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">W</font> be  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font> )</span><br/><a NAME="E1:33_1"/>
<font color="Maroon" title="c9">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="#E8:33"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="tops_3.html#T76" title="TOPS_3:th.76">TOPS_3:76</a></i>;<br/><b>hence </b><a NAME="E2:33_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font> )
 <b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="#E6:33"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="#E10:33"><i><font color="Green" title="E32">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E14:33"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> is  <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 <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow16.html#T49" title="YELLOW16:th.49">YELLOW16:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">i</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> as  <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 <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ,<font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="#E8:33"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="yellow12.html#T36" title="YELLOW12:th.36">YELLOW12:36</a></i>;<br/>
<a NAME="E16:33"/>
<font color="Maroon" title="c8">c</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">i</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> 
 
<b>by </b><i><a class="txt" href="#E8:33"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="#E10:33"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="yellow16.html#T50" title="YELLOW16:th.50">YELLOW16:50</a></i>;<br/>
<b>hence </b><a NAME="E17:33"/>
 <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="waybel25.html#D1" title="WAYBEL25:def.1">WAYBEL25:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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