<?xml version="1.0"?>
<div class="add">

<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>, <font color="Maroon">Y</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">S</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">W1</font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span>, <font color="Maroon">W2</font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font>
 ;<br/>

<b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">S</font></span>)</span>, <font color="Maroon">f2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">S</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="waybel26.html#K6" target="_self">*graph</a>  &amp; <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="waybel26.html#K6" target="_self">*graph</a>  )
 ;<br/>

<a NAME="E3:61"/><i><font color="Green">E59</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E4:61"/>
<font color="Maroon">S</font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T44">YELLOW_9:44</a></i>;<br/>
<a NAME="E5:61"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</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">S</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">S</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#T16">WAYBEL25:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">f1</font>, <font color="Maroon">g2</font> = <font color="Maroon">f2</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">S</font></span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E3:61_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/><b>reconsider </b><font color="Maroon">g1x</font> = <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>, <font color="Maroon">g2x</font> = <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/><a NAME="E5:61_1"/>
( <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:61_1"/><b>then </b>
<font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T157">RELAT_1:157</a></i>;<br/><a NAME="E7:61_1"/><b>then </b>
<font color="Maroon">g1x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g2x</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>hence </b><a NAME="E8:61_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:61"/><b>then </b>
<font color="Maroon">g1</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">g2</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="E9:61"/>
<font color="Maroon">f1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f2</font>
 <b>by </b><i/>;<br/>


</div>
