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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">S</font> be  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>;<br/><b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">S</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">f2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:45"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:45"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E5:45"/>
( <font color="Maroon" title="c4">f1</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c5">f2</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">S</font> )
 
<b>by </b><i><a class="ref" href="waybel24.html#T21" title="WAYBEL24:th.21">WAYBEL24:21</a></i>;<br/>
<a NAME="E6:45"/><b>then </b><i><font color="Green" title="E26">A4</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:45"/><i><font color="Green" title="E27">A5</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E2:45"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="waybel29.html#E4:45"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="waybel26.html#T39" title="WAYBEL26:th.39">WAYBEL26:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="waybel29.html#E2:45"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="waybel29.html#E4:45"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="waybel29.html#E6:45"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="waybel26.html#T39" title="WAYBEL26:th.39">WAYBEL26:39</a></i>;<br/>
<a NAME="E9:45"/><i><font color="Green" title="E28">A6</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c11">F2</font> = <font color="Maroon" title="c5">f2</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">S</font> <b>by </b><i><a class="ref" href="waybel24.html#T21" title="WAYBEL24:th.21">WAYBEL24:21</a></i>;<br/>
<a NAME="E11:45"/>
<font color="Maroon" title="c10">F1</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c9">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font>
 
;<br/>
<a NAME="E12:45"/><b>then </b>
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:45"><i><font color="Green" title="E28">A6</font></i></a></i>;<br/>
<a NAME="E13:45"/><b>then </b><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E14:45"/>
<font color="Maroon" title="c11">F2</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c9">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font>
 
;<br/>
<a NAME="E15:45"/><b>then </b>
<font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:45"><i><font color="Green" title="E28">A6</font></i></a></i>;<br/>
<a NAME="E16:45"/><b>then </b><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E17:45"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:45"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="waybel24.html#T20" title="WAYBEL24:th.20">WAYBEL24:20</a></i>;<br/>
<a NAME="E18:45"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:45"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E19:45"/><b>then </b>
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E13:45"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="waybel29.html#E16:45"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E20:45"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c5">f2</font>
 <b>by </b><i><a class="txt" href="waybel29.html#E4:45"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="waybel29.html#E6:45"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="waybel29.html#E7:45"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="waybel26.html#T39" title="WAYBEL26:th.39">WAYBEL26:39</a></i>;<br/>


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