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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V5">being_T4</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) ) ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="urysohn3.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = 1 as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E3:34_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 ;<br/><a NAME="E4:34_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="urysohn3.html#E3:34_1_2"><i><font color="Green">E32</font></i></a>);<br/></i><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E6:34_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>7</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E7:34_1_2"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/><b>thus </b><a NAME="E8:34_1_2"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a><br/>

<b>assume </b><a NAME="E1:34_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:34_1_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="measure5.html#K6">{}</a>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E5:34_1_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E6:34_1_2_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:34_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E8:34_1_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
;<br/>
<a NAME="E9:34_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<a NAME="E10:34_1_2_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E6:34_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_2_1_1"/>
( 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> or  not 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1"/>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E6:34_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:34_1_2_1_1_1"/>
<a href="relset_1.html#K11">K86</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>) is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="urysohn3.html#E8:34_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_2"/>
not 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure5.html#K6">{}</a>  iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E6:34_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="urysohn3.html#E7:34_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:34_1_2_1_1_2"/>
<a href="relset_1.html#K11">K86</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>) is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="urysohn3.html#E10:34_1_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E9:34_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="urysohn3.html#E6:34_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E10:34_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:34_1_2"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
