<?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">X1</font>, <font color="Maroon" title="c4">X2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>







<b>assume </b><a NAME="E1:127"/><i><font color="Green" title="E85">A1</font></i>: 
<font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c5">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font>;<br/>

<a NAME="E2:127"/><i><font color="Green" title="E86">A2</font></i>: 
( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/>
<b>hence </b><a NAME="E3:127"/>
( <font color="Maroon" title="c5">g</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="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> implies ( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</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 <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</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 <font color="Maroon" title="c4">X2</font>,<font color="Maroon" title="c2">Y</font> ) )
 <b>by </b><i><a class="ref" href="tmap_1.html#T89" target="_self" title="TMAP_1:th.89">Th89</a></i>;<br/>

<b>thus </b><a NAME="E4:127"/>
( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</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 <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</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 <font color="Maroon" title="c4">X2</font>,<font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c5">g</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="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:127_1"/><i><font color="Green" title="E87">A3</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</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 <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</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 <font color="Maroon" title="c4">X2</font>,<font color="Maroon" title="c2">Y</font> )
 ;<br/>

<a NAME="E2:127_1"/>
for <font color="Olive" title="b1">x</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="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> holds  <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>;<br/>

<b>set </b><font color="Maroon" title="c7">X0</font> = <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>;<br/>
<a NAME="E1:127_1_1"/><i><font color="Green" title="E88">A4</font></i>: 
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:127_1_1_1"/>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font>
 ;<br/>

<a NAME="E2:127_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:127"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="tsep_1.html#T85" title="TSEP_1:th.85">TSEP_1:85</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">Y1</font>, <font color="Maroon" title="c9">Y2</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:127_1_1_1"/><i><font color="Green" title="E89">A5</font></i>: 
( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> &amp; <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c9">Y2</font> )
 <b>and </b><br/><a NAME="E5:127_1_1_1"/><i><font color="Green" title="E90">A6</font></i>: 
( <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c9">Y2</font> or <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T83" title="TSEP_1:th.83">TSEP_1:83</a></i>;<br/>
<a NAME="E6:127_1_1_1"/>
( <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c3">X1</font> )
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:127_1_1_1"><i><font color="Green" title="E89">A5</font></i></a>, <a class="txt" href="tmap_1.html#E5:127_1_1_1"><i><font color="Green" title="E90">A6</font></i></a>, <a class="ref" href="tmap_1.html#T35" target="_self" title="TMAP_1:th.35">Th35</a></i>;<br/>
<a NAME="E7:127_1_1_1"/><b>then </b><i><font color="Green" title="E91">A7</font></i>: 
( <font color="Maroon" title="c3">X1</font> is  <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c4">X2</font> is  <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:127_1_1_1"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="tmap_1.html#T46" target="_self" title="TMAP_1:th.46">Th46</a></i>;<br/>
<div><i><font color="Green" title="E92">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font><b> such that </b><a NAME="E2:127_1_1_1_1"/><i><font color="Green" title="E93">A9</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><a NAME="E3:127_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c10">x1</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E4:127_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E7:127_1_1_1"><i><font color="Green" title="E91">A7</font></i></a>, <a class="txt" href="tmap_1.html#E2:127_1_1_1_1"><i><font color="Green" title="E93">A9</font></i></a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X2</font><b> such that </b><a NAME="E2:127_1_1_1_2"/><i><font color="Green" title="E93">A10</font></i>: 
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><a NAME="E3:127_1_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c10">x2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E4:127_1_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E7:127_1_1_1"><i><font color="Green" title="E91">A7</font></i></a>, <a class="txt" href="tmap_1.html#E2:127_1_1_1_2"><i><font color="Green" title="E93">A10</font></i></a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:127_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E8:127_1_1_1"><i><font color="Green" title="E92">A8</font></i></a>, <a class="ref" href="tmap_1.html#T16" target="_self" title="TMAP_1:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:127_1_1"/>
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:127_1_1_2"/><i><font color="Green" title="E89">A11</font></i>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c4">X2</font>
 ;<br/>

<div><i><font color="Green" title="E90">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:127_1_1_2_1"/><i><font color="Green" title="E91">A13</font></i>: 
( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> or <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> )
 ;<br/><div><i><font color="Green" title="E92">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:127_1_1_2_1_1"/>
<font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font>
 ;<br/><a NAME="E2:127_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E93">A15</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">X2</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c6">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X2</font> ;<br/><a NAME="E4:127_1_1_2_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E5:127_1_1_2_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127_1_1_2_1_1"><i><font color="Green" title="E93">A15</font></i></a>, <a class="ref" href="tmap_1.html#T88" target="_self" title="TMAP_1:th.88">Th88</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:127_1_1_2_1_2"/>
<font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font>
 ;<br/><a NAME="E2:127_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E93">A16</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">X1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font> ;<br/><a NAME="E4:127_1_1_2_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x1</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E5:127_1_1_2_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127_1_1_2_1_2"><i><font color="Green" title="E93">A16</font></i></a>, <a class="ref" href="tmap_1.html#T88" target="_self" title="TMAP_1:th.88">Th88</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:127_1_1_2_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="txt" href="tmap_1.html#E2:127_1_1_2_1"><i><font color="Green" title="E92">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:127_1_1_2_2"/><i><font color="Green" title="E91">A17</font></i>: 
( <font color="Maroon" title="c3">X1</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c4">X2</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c8">Y1</font>, <font color="Maroon" title="c9">Y2</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:127_1_1_2_2"/><i><font color="Green" title="E92">A18</font></i>: 
<font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font>
 <b>and </b><br/><a NAME="E4:127_1_1_2_2"/><i><font color="Green" title="E93">A19</font></i>: 
<font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font>
 <b>and </b><br/><a NAME="E5:127_1_1_2_2"/><i><font color="Green" title="E94">A20</font></i>: 
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font> or  ex <font color="Olive" title="b1">Z</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b1">Z</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> ) )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A11</font></i></a>, <a class="ref" href="tsep_1.html#T96" title="TSEP_1:th.96">TSEP_1:96</a></i>;<br/><a NAME="E6:127_1_1_2_2"/><i><font color="Green" title="E95">A21</font></i>: 
( <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is  <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self" title="TMAP_1:th.44">Th44</a></i>;<br/><a NAME="E7:127_1_1_2_2"/><i><font color="Green" title="E96">A22</font></i>: 
( <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is  <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self" title="TMAP_1:th.44">Th44</a></i>;<br/><div><i><font color="Green" title="E97">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:127_1_1_2_2_1"/><i><font color="Green" title="E98">A24</font></i>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font>
 ;<br/><a NAME="E2:127_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E99">A25</font></i>: 
<font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="ref" href="tmap_1.html#T36" target="_self" title="TMAP_1:th.36">Th36</a></i>;<br/><a NAME="E3:127_1_1_2_2_1"/><i><font color="Green" title="E100">A26</font></i>: 
<font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2_1"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="tmap_1.html#T36" target="_self" title="TMAP_1:th.36">Th36</a></i>;<br/><a NAME="E4:127_1_1_2_2_1"/>
<font color="Maroon" title="c8">Y1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/><a NAME="E5:127_1_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127_1_1_2_2_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="ref" href="tmap_1.html#T23" target="_self" title="TMAP_1:th.23">Th23</a></i>;<br/><a NAME="E6:127_1_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2_2_1"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="tsep_1.html#T31" title="TSEP_1:th.31">TSEP_1:31</a></i>;<br/><a NAME="E7:127_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E101">A27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127_1_1_2_2_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2_1"><i><font color="Green" title="E100">A26</font></i></a>, <a class="ref" href="tsep_1.html#T35" title="TSEP_1:th.35">TSEP_1:35</a></i>;<br/><div><i><font color="Green" title="E102">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_2_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">x1</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="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E2:127_1_1_2_2_1_1"/><i><font color="Green" title="E103">A29</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><a NAME="E3:127_1_1_2_2_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="ref" href="tmap_1.html#T90" target="_self" title="TMAP_1:th.90">Th90</a></i>;<br/><a NAME="E4:127_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c10">x1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E5:127_1_1_2_2_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="tmap_1.html#E6:127_1_1_2_2"><i><font color="Green" title="E95">A21</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2_1"><i><font color="Green" title="E98">A24</font></i></a>, <a class="txt" href="tmap_1.html#E2:127_1_1_2_2_1_1"><i><font color="Green" title="E103">A29</font></i></a>, <a class="ref" href="tmap_1.html#T36" target="_self" title="TMAP_1:th.36">Th36</a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_2_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">x2</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="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E2:127_1_1_2_2_1_2"/><i><font color="Green" title="E103">A30</font></i>: 
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><a NAME="E3:127_1_1_2_2_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="ref" href="tmap_1.html#T90" target="_self" title="TMAP_1:th.90">Th90</a></i>;<br/><a NAME="E4:127_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c10">x2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E5:127_1_1_2_2_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="tmap_1.html#E7:127_1_1_2_2"><i><font color="Green" title="E96">A22</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2_1"><i><font color="Green" title="E98">A24</font></i></a>, <a class="txt" href="tmap_1.html#E2:127_1_1_2_2_1_2"><i><font color="Green" title="E103">A30</font></i></a>, <a class="ref" href="tmap_1.html#T36" target="_self" title="TMAP_1:th.36">Th36</a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:127_1_1_2_2_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E7:127_1_1_2_2_1"><i><font color="Green" title="E101">A27</font></i></a>, <a class="txt" href="tmap_1.html#E8:127_1_1_2_2_1"><i><font color="Green" title="E102">A28</font></i></a>, <a class="ref" href="tmap_1.html#T16" target="_self" title="TMAP_1:th.16">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:127_1_1_2_2_2"/><i><font color="Green" title="E98">A31</font></i>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">Z</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:127_1_1_2_2_2"/><i><font color="Green" title="E99">A32</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c10">Z</font>
 <b>and </b><br/><a NAME="E4:127_1_1_2_2_2"/><i><font color="Green" title="E100">A33</font></i>: 
<font color="Maroon" title="c10">Z</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E5:127_1_1_2_2"><i><font color="Green" title="E94">A20</font></i></a></i>;<br/><a NAME="E5:127_1_1_2_2_2"/><i><font color="Green" title="E101">A34</font></i>: 
<font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2_2"><i><font color="Green" title="E98">A31</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2_2"><i><font color="Green" title="E99">A32</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2_2"><i><font color="Green" title="E100">A33</font></i></a>, <a class="ref" href="tmap_1.html#T38" target="_self" title="TMAP_1:th.38">Th38</a></i>;<br/><a NAME="E6:127_1_1_2_2_2"/><i><font color="Green" title="E102">A35</font></i>: 
<font color="Maroon" title="c10">Z</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2_2"><i><font color="Green" title="E98">A31</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2_2"><i><font color="Green" title="E99">A32</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2_2"><i><font color="Green" title="E100">A33</font></i></a>, <a class="ref" href="tmap_1.html#T38" target="_self" title="TMAP_1:th.38">Th38</a></i>;<br/><a NAME="E7:127_1_1_2_2_2"/><i><font color="Green" title="E103">A36</font></i>: 
<font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2_2"><i><font color="Green" title="E99">A32</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2_2"><i><font color="Green" title="E100">A33</font></i></a>, <a class="ref" href="tmap_1.html#T37" target="_self" title="TMAP_1:th.37">Th37</a></i>;<br/><a NAME="E8:127_1_1_2_2_2"/><i><font color="Green" title="E104">A37</font></i>: 
<font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2_2"><i><font color="Green" title="E99">A32</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2_2"><i><font color="Green" title="E100">A33</font></i></a>, <a class="ref" href="tmap_1.html#T37" target="_self" title="TMAP_1:th.37">Th37</a></i>;<br/><a NAME="E9:127_1_1_2_2_2"/>
<font color="Maroon" title="c1">X</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">X12</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> #) as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tmap_1.html#T11" target="_self" title="TMAP_1:th.11">Th11</a></i>;<br/><a NAME="E11:127_1_1_2_2_2"/>
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="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">X12</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T35" title="BORSUK_1:th.35">BORSUK_1:35</a></i>;<br/><a NAME="E12:127_1_1_2_2_2"/><b>then </b><i><font color="Green" title="E105">A38</font></i>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c11">X12</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/><a NAME="E13:127_1_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c11">X12</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T22" target="_self" title="TMAP_1:th.22">Th22</a></i>;<br/><a NAME="E14:127_1_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c10">Z</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 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="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> #)
 <b>by </b><i><a class="txt" href="tmap_1.html#E3:127_1_1_2_2_2"><i><font color="Green" title="E99">A32</font></i></a>, <a class="txt" href="tmap_1.html#E12:127_1_1_2_2_2"><i><font color="Green" title="E105">A38</font></i></a>, <a class="ref" href="tsep_1.html#T31" title="TSEP_1:th.31">TSEP_1:31</a></i>;<br/><a NAME="E15:127_1_1_2_2_2"/><b>then </b><i><font color="Green" title="E106">A39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# 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="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> #)
 <b>by </b><i><a class="txt" href="tmap_1.html#E5:127_1_1_2_2_2"><i><font color="Green" title="E101">A34</font></i></a>, <a class="txt" href="tmap_1.html#E6:127_1_1_2_2_2"><i><font color="Green" title="E102">A35</font></i></a>, <a class="ref" href="tsep_1.html#T35" title="TSEP_1:th.35">TSEP_1:35</a></i>;<br/><div><i><font color="Green" title="E107">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:127_1_1_2_2_2_1"/><i><font color="Green" title="E108">A41</font></i>: 
 ex <font color="Olive" title="b1">x12</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> st <font color="Olive" title="b1">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><a NAME="E2:127_1_1_2_2_2_1"/><i><font color="Green" title="E109">A42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E7:127_1_1_2_2_2"><i><font color="Green" title="E103">A36</font></i></a>, <a class="txt" href="tmap_1.html#E8:127_1_1_2_2_2"><i><font color="Green" title="E104">A37</font></i></a>, <a class="ref" href="tsep_1.html#T35" title="TSEP_1:th.35">TSEP_1:35</a></i>;<br/><div><i><font color="Green" title="E110">A43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_2_2_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c12">x1</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="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E2:127_1_1_2_2_2_1_1"/><i><font color="Green" title="E111">A44</font></i>: 
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><a NAME="E3:127_1_1_2_2_2_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="ref" href="tmap_1.html#T90" target="_self" title="TMAP_1:th.90">Th90</a></i>;<br/><a NAME="E4:127_1_1_2_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c12">x1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E5:127_1_1_2_2_2_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="tmap_1.html#E6:127_1_1_2_2"><i><font color="Green" title="E95">A21</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2_2"><i><font color="Green" title="E99">A32</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2_2"><i><font color="Green" title="E100">A33</font></i></a>, <a class="txt" href="tmap_1.html#E2:127_1_1_2_2_2_1_1"><i><font color="Green" title="E111">A44</font></i></a>, <a class="ref" href="tmap_1.html#T37" target="_self" title="TMAP_1:th.37">Th37</a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_2_2_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c12">x2</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="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E2:127_1_1_2_2_2_1_2"/><i><font color="Green" title="E111">A45</font></i>: 
<font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><a NAME="E3:127_1_1_2_2_2_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="ref" href="tmap_1.html#T90" target="_self" title="TMAP_1:th.90">Th90</a></i>;<br/><a NAME="E4:127_1_1_2_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c12">x2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E5:127_1_1_2_2_2_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="tmap_1.html#E1:127_1_1_2_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2"><i><font color="Green" title="E93">A19</font></i></a>, <a class="txt" href="tmap_1.html#E7:127_1_1_2_2"><i><font color="Green" title="E96">A22</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2_2"><i><font color="Green" title="E99">A32</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2_2"><i><font color="Green" title="E100">A33</font></i></a>, <a class="txt" href="tmap_1.html#E2:127_1_1_2_2_2_1_2"><i><font color="Green" title="E111">A45</font></i></a>, <a class="ref" href="tmap_1.html#T37" target="_self" title="TMAP_1:th.37">Th37</a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:127_1_1_2_2_2_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2_2_2_1"><i><font color="Green" title="E108">A41</font></i></a>, <a class="txt" href="tmap_1.html#E2:127_1_1_2_2_2_1"><i><font color="Green" title="E109">A42</font></i></a>, <a class="txt" href="tmap_1.html#E3:127_1_1_2_2_2_1"><i><font color="Green" title="E110">A43</font></i></a>, <a class="ref" href="tmap_1.html#T16" target="_self" title="TMAP_1:th.16">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_2_2_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c12">x0</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="c10">Z</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E2:127_1_1_2_2_2_2"/><i><font color="Green" title="E108">A46</font></i>: 
<font color="Maroon" title="c12">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/><b>consider </b><font color="Maroon" title="c13">x00</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="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span><b> such that </b><br/><a NAME="E4:127_1_1_2_2_2_2"/><i><font color="Green" title="E109">A47</font></i>: 
<font color="Maroon" title="c13">x00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x0</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E4:127_1_1_2_2_2"><i><font color="Green" title="E100">A33</font></i></a>, <a class="ref" href="tmap_1.html#T15" target="_self" title="TMAP_1:th.15">Th15</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font><b> such that </b><br/><a NAME="E6:127_1_1_2_2_2_2"/><i><font color="Green" title="E110">A48</font></i>: 
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x00</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A11</font></i></a>, <a class="ref" href="tmap_1.html#T17" target="_self" title="TMAP_1:th.17">Th17</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X2</font><b> such that </b><br/><a NAME="E8:127_1_1_2_2_2_2"/><i><font color="Green" title="E111">A49</font></i>: 
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x00</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A11</font></i></a>, <a class="ref" href="tmap_1.html#T17" target="_self" title="TMAP_1:th.17">Th17</a></i>;<br/><a NAME="E9:127_1_1_2_2_2_2"/>
( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c14">x1</font> &amp; <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c15">x2</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E10:127_1_1_2_2_2_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127_1_1_2_2_2_2"><i><font color="Green" title="E108">A46</font></i></a>, <a class="txt" href="tmap_1.html#E4:127_1_1_2_2_2_2"><i><font color="Green" title="E109">A47</font></i></a>, <a class="txt" href="tmap_1.html#E6:127_1_1_2_2_2_2"><i><font color="Green" title="E110">A48</font></i></a>, <a class="txt" href="tmap_1.html#E8:127_1_1_2_2_2_2"><i><font color="Green" title="E111">A49</font></i></a>, <a class="ref" href="tmap_1.html#T123" target="_self" title="TMAP_1:th.123">Th123</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:127_1_1_2_2_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E15:127_1_1_2_2_2"><i><font color="Green" title="E106">A39</font></i></a>, <a class="txt" href="tmap_1.html#E16:127_1_1_2_2_2"><i><font color="Green" title="E107">A40</font></i></a>, <a class="ref" href="tmap_1.html#T16" target="_self" title="TMAP_1:th.16">Th16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:127_1_1_2_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E8:127_1_1_2_2"><i><font color="Green" title="E97">A23</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:127_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:127_1_1_2"><i><font color="Green" title="E90">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:127_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:127_1_1"><i><font color="Green" title="E88">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:127_1"/>
<font color="Maroon" title="c5">g</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="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/>


</div><b>end;</b></div>


</div>
