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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a> <a href="yellow13.html#V2" target="_self">topological_semilattice</a>  <a href="waybel_9.html#L1">TopRelStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a><br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>;<br/>







<b>set </b><font color="Maroon">TT</font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>
<a NAME="E1:66"/><i><font color="Green">E28</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E2:66"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">T</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E4:66"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">G</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xp</font> = <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">p</font></span><a href="yellow_3.html#K7">]</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, <a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<i><font color="Green">E36</font></i>: <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font> = 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>

<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="waybel_2.html#D4">WAYBEL_2:def 4</a></i>
;<br/>
<a NAME="E7:66"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">xp</font><b> such that </b><br/><a NAME="E9:66"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E11:66"/><i><font color="Green">E39</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E12:66"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">X1</font>, <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="yellow_3.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<a NAME="E13:66"/>
<font color="Maroon">xp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:66"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">xp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E16:66"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">X1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">Y1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E18:66"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="yellow_3.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E19:66"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E20:66"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 
<b>by </b><i>, <a class="ref" href="yellow13.html#T1" target="_self">Th1</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Y1</font> = <font color="Maroon">Y1</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font> <b>by </b><i>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>take </b>
<font color="Maroon">Y1</font>
;<br/>

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E22:66"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:66"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span>
 <b>and </b><br/><a NAME="E25:66"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>and </b><br/><a NAME="E26:66"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="yellow13.html#E4:66"><i><font color="Green">E35</font></i></a></i>;<br/>
<i><font color="Green">E108</font></i>: <font color="Maroon">b</font> = 
<font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="yellow13.html#E6:66"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="waybel_2.html#D4">WAYBEL_2:def 4</a></i>
;<br/>
<a NAME="E29:66"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, <a class="ref" href="yellow13.html#T1" target="_self">Th1</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E30:66"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="yellow13.html#T1" target="_self">Th1</a>, <a class="txt" href="yellow13.html#E4:66"><i><font color="Green">E35</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E31:66"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E32:66"/><i><font color="Green">E110</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E33:66"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E34:66"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E35:66"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="txt" href="yellow13.html#E9:66"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yellow13.html#E11:66"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E36:66"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="yellow13.html#E1:66"><i><font color="Green">E28</font></i></a></i>;<br/>


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