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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">p0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <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="c2">Y</font>;<br/>









<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
( <font color="Maroon" title="c4">D</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p0</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c1">X</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a>  &amp; <font color="Maroon" title="c2">Y</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a>  &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D</font></span>)</span>  holds <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p0</font> ) &amp; <font color="Maroon" title="c6">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">D</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">E</font></span>)</span> &amp; ( for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> ) ) )
 ;<br/>

<a NAME="E2:3"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c8">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>;<br/>



<b>assume </b><a NAME="E1:3_1"/><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> &amp; <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>

<a NAME="E2:3_1"/><i><font color="Green" title="E6">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1"/>
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p0</font> or <font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p0</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">V</font> )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="jgraph_3.html#E1:3_1"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_2"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p0</font>
 ;<br/><div class="add"><a NAME="E2:3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:3_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">p</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 <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:3_1_1_2"/><b>then </b><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></i>;<br/><a NAME="E5:3_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p0</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="jgraph_3.html#E2:3_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">G1</font>, <font color="Maroon" title="c10">G2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E7:3_1_1_2"/><i><font color="Green" title="E9">A6</font></i>: 
( <font color="Maroon" title="c9">G1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">G2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">G1</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G2</font> &amp; <font color="Maroon" title="c9">G1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">G2</font> )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="compts_1.html#D4" title="COMPTS_1:def.4">COMPTS_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">E</font></span>)</span><b> such that </b><br/><a NAME="E9:3_1_1_2"/><i><font color="Green" title="E10">A7</font></i>: 
( <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c11">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/><a NAME="E10:3_1_1_2"/><i><font color="Green" title="E11">A8</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">p22</font> = <font color="Maroon" title="c7">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D</font></span>)</span> <b>by </b><i><a class="txt" href="jgraph_3.html#E4:3_1_1_2"><i><font color="Green" title="E8">A5</font></i></a></i>;<br/><a NAME="E12:3_1_1_2"/><i><font color="Green" title="E12">A9</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">V20</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">G1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">V</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">E</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E14:3_1_1_2"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E4:3_1_1_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="jgraph_3.html#E9:3_1_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E15:3_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p0</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="jgraph_3.html#E4:3_1_1_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="jgraph_3.html#E10:3_1_1_2"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/><a NAME="E16:3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E17:3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</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 <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></i>;<br/><a NAME="E18:3_1_1_2"/><b>then </b><i><font color="Green" title="E14">A11</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">p22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E14:3_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E19:3_1_1_2"/>
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">p22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">G1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="jgraph_3.html#E7:3_1_1_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="jgraph_3.html#E14:3_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E20:3_1_1_2"/><b>then </b><i><font color="Green" title="E15">A12</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">p22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">V20</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E18:3_1_1_2"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E21:3_1_1_2"/>
<font color="Maroon" title="c9">G1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="jgraph_3.html#E7:3_1_1_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/><a NAME="E22:3_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">V20</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E12:3_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">W2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D</font></span>)</span><b> such that </b><br/><a NAME="E24:3_1_1_2"/><i><font color="Green" title="E16">A13</font></i>: 
( <font color="Maroon" title="c12">p22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W2</font> &amp; <font color="Maroon" title="c14">W2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c11">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">V20</font> )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E9:3_1_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="jgraph_3.html#E20:3_1_1_2"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">W3b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E26:3_1_1_2"/><i><font color="Green" title="E17">A14</font></i>: 
( <font color="Maroon" title="c15">W3b</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c14">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">W3b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E24:3_1_1_2"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/><a NAME="E27:3_1_1_2"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c12">p22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">W3b</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E24:3_1_1_2"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="jgraph_3.html#E26:3_1_1_2"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">H1</font>, <font color="Maroon" title="c17">H2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E29:3_1_1_2"/><i><font color="Green" title="E19">A16</font></i>: 
( <font color="Maroon" title="c16">H1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c17">H2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H1</font> &amp; <font color="Maroon" title="c3">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H2</font> &amp; <font color="Maroon" title="c16">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">H2</font> )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="jgraph_3.html#E1:3_1_1_2"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="compts_1.html#D4" title="COMPTS_1:def.4">COMPTS_1:def 4</a></i>;<br/><a NAME="E30:3_1_1_2"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c16">H1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">W3b</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E26:3_1_1_2"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="jgraph_3.html#E29:3_1_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/><a NAME="E31:3_1_1_2"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">W3b</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E27:3_1_1_2"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="jgraph_3.html#E29:3_1_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">W3</font> = <font color="Maroon" title="c16">H1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">W3b</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E33:3_1_1_2"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c18">W3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">W3b</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E34:3_1_1_2"/><i><font color="Green" title="E23">A20</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c18">W3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">W2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">xx</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="E1:3_1_1_2_1"/>
<font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c18">W3</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">yy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_1_1_2_1"/><i><font color="Green" title="E24">A21</font></i>: 
( <font color="Maroon" title="c20">yy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c20">yy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">W3</font> &amp; <font color="Maroon" title="c19">xx</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="c20">yy</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="E4:3_1_1_2_1"/><i><font color="Green" title="E25">A22</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E9:3_1_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E5:3_1_1_2_1"/><i><font color="Green" title="E26">A23</font></i>: 
<font color="Maroon" title="c18">W3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">H1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:3_1_1_2_1"/>
<font color="Maroon" title="c17">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">H1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E29:3_1_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<a NAME="E7:3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">H1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="jgraph_3.html#E29:3_1_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c16">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></i>;<br/>
<a NAME="E9:3_1_1_2_1"/><b>then </b><i><font color="Green" title="E27">A24</font></i>: 
<font color="Maroon" title="c18">W3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E5:3_1_1_2_1"><i><font color="Green" title="E26">A23</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:3_1_1_2_1"/><b>then </b><i><font color="Green" title="E28">A25</font></i>: 
<font color="Maroon" title="c20">yy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">h</font>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E3:3_1_1_2_1"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="jgraph_3.html#E4:3_1_1_2_1"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:3_1_1_2_1"/><b>then </b><i><font color="Green" title="E29">A26</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">yy</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="c20">yy</font>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E9:3_1_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E12:3_1_1_2_1"/>
<font color="Maroon" title="c20">yy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W2</font>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E10:3_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="jgraph_3.html#E26:3_1_1_2"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="jgraph_3.html#E33:3_1_1_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="jgraph_3.html#E3:3_1_1_2_1"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="jgraph_3.html#E9:3_1_1_2_1"><i><font color="Green" title="E27">A24</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:3_1_1_2_1"/>
<font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">W2</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E3:3_1_1_2_1"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="jgraph_3.html#E10:3_1_1_2_1"><i><font color="Green" title="E28">A25</font></i></a>, <a class="txt" href="jgraph_3.html#E11:3_1_1_2_1"><i><font color="Green" title="E29">A26</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><a NAME="E35:3_1_1_2"/><i><font color="Green" title="E24">A27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">G1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">V</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E36:3_1_1_2"/><i><font color="Green" title="E25">A28</font></i>: 
<font color="Maroon" title="c9">G1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E37:3_1_1_2"/>
<font color="Maroon" title="c11">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E24:3_1_1_2"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="jgraph_3.html#E35:3_1_1_2"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E38:3_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E36:3_1_1_2"><i><font color="Green" title="E25">A28</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E39:3_1_1_2"/>
 ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">V</font> )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E30:3_1_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="jgraph_3.html#E31:3_1_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="jgraph_3.html#E34:3_1_1_2"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
<font color="Maroon" title="c6">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>


</div>
