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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="realset2.html#V3">trivial</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/>



<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:35"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>assume </b><a NAME="E3:35"/>
 <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is <a href="lattice3.html#V1">with_suprema</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">XY</font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> as   <a href="waybel_0.html#NM7">sup-Semilattice</a> ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">X</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">a</font>, <font color="Maroon">g</font> = <font color="Maroon">X</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">b</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">ef</font> = <font color="Maroon">f</font>, <font color="Maroon">eg</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">XY</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">ef</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">eg</font>, <font color="Maroon">f</font> = <font color="Maroon">ef</font>, <font color="Maroon">g</font> = <font color="Maroon">eg</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E9:35"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span>  holds <br/>(  not <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> or  not <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> )
 ;<br/><a NAME="E2:35_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( (  not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> or  not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> ) &amp; (  not <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> or  not <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> ) )
 ;<br/><a NAME="E3:35_1"/>
( <font color="Maroon">ef</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">ef</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">eg</font> &amp; <font color="Maroon">eg</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">ef</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">eg</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E4:35_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">g</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">h</font> )
 <b>by </b><i/>;<br/><a NAME="E5:35_1"/><b>then </b>
(  ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> ) &amp;  ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> ) )
 <b>by </b><i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/><b>hence </b><a NAME="E6:35_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span><b> such that </b><br/><a NAME="E12:35"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>
<a NAME="E13:35"/><i><font color="Green">E62</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#D2">WAYBEL25:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font>, <font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<b>take </b>
<font color="Maroon">p</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="urysohn1.html#D9">URYSOHN1:def 9</a><br/>

<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E15:35"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E16:35"/>
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i>, <a class="ref" href="tops_3.html#T76">TOPS_3:76</a></i>;<br/>
<a NAME="E18:35"/>
<font color="Maroon">W</font> is <a href="waybel_0.html#V13">upper</a>
 
;<br/>
<b>hence </b><a NAME="E19:35"/>
(  not <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> or  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> or  not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>


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