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<span class="kw">let </span><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" 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"> (  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/>

<span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E2:32"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:32"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp;  not <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">V</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">V</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="urysohn1.html#D7" onmouseover="rs('urysohn1/D7')" onmouseout="rh()">URYSOHN1:def 7</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="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="E4:32"/>
(  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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp;  not <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="tsp_1.html#D3" onmouseover="rs('tsp_1/D3')" onmouseout="rh()">TSP_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:32"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c4">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E7:32"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
( ( <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp;  not <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">V</font> ) or (  not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">V</font> ) )
 ;<br/>
<a NAME="E8:32"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
 <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> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel25.html#D2" onmouseover="rs('waybel25/D2')" onmouseout="rh()">WAYBEL25:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E10:32"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
( ( <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp;  not <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI: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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">now </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">  for <font color="Olive" title="b1">W</font> being   <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>  st <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font> holds <br/><font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">W</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">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" 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"> ( <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">W</font> implies <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">W</font> )</span><br/><a NAME="E1:32_1"/>
<font color="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="tops_3.html#T76" onmouseover="rs('tops_3/T76')" onmouseout="rh()">TOPS_3:76</a></span>;<br/><span class="kw">hence </span><a NAME="E2:32_1"/>
( <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">W</font> implies <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E32">A6</font></span></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/></div><span class="kw">end;</span></div>
<a NAME="E12:32"/><span class="kw">then </span>
(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow16.html#T47" onmouseover="rs('yellow16/T47')" onmouseout="rh()">YELLOW16:47</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">i</font> = (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="yellow12.html#T36" onmouseover="rs('yellow12/T36')" onmouseout="rh()">YELLOW12:36</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E8:32"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="tops_3.html#T76" onmouseover="rs('tops_3/T76')" onmouseout="rh()">TOPS_3:76</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">c</font> =  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c8">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="yellow16.html#T46" onmouseover="rs('yellow16/T46')" onmouseout="rh()">YELLOW16:46</a></span>;<br/>
<a NAME="E16:32"/>
<font color="Maroon" title="c9">c</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="yellow16.html#T48" onmouseover="rs('yellow16/T48')" onmouseout="rh()">YELLOW16:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:32"/>
 <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 class="kw">by</span> <span class="lab"><a class="ref" href="waybel25.html#D1" onmouseover="rs('waybel25/D1')" onmouseout="rh()">WAYBEL25:def 1</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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