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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>, <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">f1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</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="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">S</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f2</font>
 ;<br/>

<b>let </b><font color="Maroon">x</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="E2:45"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel26.html#K5">*graph</a> <font color="Maroon">f1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:45"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E5:45"/>
( <font color="Maroon">f1</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">S</font> &amp; <font color="Maroon">f2</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="waybel24.html#T21">WAYBEL24:21</a></i>;<br/>
<a NAME="E6:45"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:45"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="waybel26.html#T39">WAYBEL26:39</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">X</font> <b>by </b><i>, , , <a class="ref" href="waybel26.html#T39">WAYBEL26:39</a></i>;<br/>
<a NAME="E9:45"/><i><font color="Green">E54</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/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">f1</font>, <font color="Maroon">F2</font> = <font color="Maroon">f2</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel24.html#T21">WAYBEL24:21</a></i>;<br/>
<a NAME="E11:45"/>
<font color="Maroon">F1</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E12:45"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</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>
 
<b>by </b><i/>;<br/>
<a NAME="E13:45"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E14:45"/>
<font color="Maroon">F2</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E15:45"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</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>
 
<b>by </b><i/>;<br/>
<a NAME="E16:45"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E17:45"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="waybel24.html#T20">WAYBEL24:20</a></i>;<br/>
<a NAME="E18:45"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E19:45"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E20:45"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel26.html#K5">*graph</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , , <a class="ref" href="waybel26.html#T39">WAYBEL26:39</a></i>;<br/>


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