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

<b>let </b><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   <a href="coh_sp.html#NM1" title="COH_SP:NM.1">Coherence_Space</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be  <a href="cohsp_1.html#V8" title="COHSP_1:attr.8">U-continuous</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>;<br/>



<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E35">A1</font></i>: 
 <a href="cohsp_1.html#K6" title="COHSP_1:func.6">graph</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K6" title="COHSP_1:func.6">graph</a> <font color="Maroon" title="c4">g</font>
 ;<br/>

<a NAME="E2:70"/><i><font color="Green" title="E36">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</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/>
<div><i><font color="Green" title="E37">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</font> be  <a href="cohsp_1.html#V8" title="COHSP_1:attr.8">U-continuous</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>;<br/><b>assume </b><a NAME="E1:70_1"/><i><font color="Green" title="E38">A4</font></i>: 
 <a href="cohsp_1.html#K6" title="COHSP_1:func.6">graph</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K6" title="COHSP_1:func.6">graph</a> <font color="Maroon" title="c7">g</font>
 ;<br/><a NAME="E2:70_1"/><i><font color="Green" title="E39">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</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/><b>thus </b><a NAME="E3:70_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">z</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="E1:70_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 ;<br/>

<a NAME="E2:70_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c8">z</font></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="cohsp_1.html#K6" title="COHSP_1:func.6">graph</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E2:70_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="cohsp_1.html#T25" target="_self" title="COHSP_1:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E3:70_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:70_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="cohsp_1.html#T25" target="_self" title="COHSP_1:th.25">Th25</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E38">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</font> be  <a href="cohsp_1.html#V8" title="COHSP_1:attr.8">U-continuous</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>;<br/><b>assume </b><a NAME="E1:70_2"/><i><font color="Green" title="E39">A7</font></i>: 
 <a href="cohsp_1.html#K6" title="COHSP_1:func.6">graph</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K6" title="COHSP_1:func.6">graph</a> <font color="Maroon" title="c7">g</font>
 ;<br/><a NAME="E2:70_2"/><i><font color="Green" title="E40">A8</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</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/><b>thus </b><a NAME="E3:70_2"/>
<font color="Maroon" title="c6">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</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="E1:70_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:70_2_1"/><i><font color="Green" title="E41">A9</font></i>: 
( <font color="Maroon" title="c9">x</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="c6">f</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:70_2_1"/>
<font color="Maroon" title="c9">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E3:70_2_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E5:70_2_1"/><b>then </b>
( <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E3:70"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="cohsp_1.html#E1:70_2"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="cohsp_1.html#E3:70_2_1"><i><font color="Green" title="E41">A9</font></i></a></i>;<br/>
<a NAME="E6:70_2_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:70_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E2:70_2"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="cohsp_1.html#E3:70_2_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>;<br/><a NAME="E1:70_3"/>
( <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span> &amp; <font color="Maroon" title="c4">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:70"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="cohsp_1.html#E4:70"><i><font color="Green" title="E38">A6</font></i></a></i>;<br/><a NAME="E2:70_3"/><b>then </b><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>thus </b><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cohsp_1.html#E2:70"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="cohsp_1.html#T21" target="_self" title="COHSP_1:th.21">Th21</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="cohsp_1.html#E2:70"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="cohsp_1.html#E2:70_3"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="cohsp_1.html#T21" target="_self" title="COHSP_1:th.21">Th21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:70"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
