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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</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  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:158"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3">are_separated</a>  implies ( <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Olive">b<sub>3</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:158_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3">are_separated</a> 
 ;<br/>

<a NAME="E2:158_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R4">are_weakly_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T85">TSEP_1:85</a></i>;<br/>
<b>thus </b><a NAME="E3:158_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:158_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="tsep_1.html#T68">TSEP_1:68</a></i>;<br/>

<b>hence </b><a NAME="E4:158_1"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Olive">b<sub>3</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:158_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="tmap_1.html#T145" target="_self">Th145</a></i>;<br/>


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

<b>thus </b><a NAME="E2:158"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Olive">b<sub>3</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:158_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:158_2"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Olive">b<sub>3</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:158_2_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>2</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="tmap_1.html#E1:158_2_1"><i><font color="Green">E109</font></i></a></i>;<br/><a NAME="E4:158_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T138" target="_self">Th138</a></i>;<br/><b>hence </b><a NAME="E5:158_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:158_2"><i><font color="Green">E108</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:158_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3">are_separated</a> 
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:158_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="tmap_1.html#T135" target="_self">Th135</a></i>;<br/>


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


</div>
