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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>3</sub></font>, <font color="Maroon">c<sub>4</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>assume </b><a NAME="E1:148"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>thus </b><a NAME="E2:148"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1"><b>proof </b></a><div class="add">

<a NAME="E1:148_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:148"><i><font color="Green">E106</font></i></a>, <a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/>
<a NAME="E2:148_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>assume </b><a NAME="E3:148_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E4:148_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:148_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="tmap_1.html#E2:148_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="tmap_1.html#T79" target="_self">Th79</a></i>;<br/>
<b>hence </b><a NAME="E5:148_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>


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

<b>thus </b><a NAME="E3:148"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> implies ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> ) )
 <b>by </b><i><a class="ref" href="tmap_1.html#D12" target="_self">Def12</a></i>;<br/>


</div>
