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

<b>let </b><font color="Maroon">X</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">Y</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">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">X2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">X3</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">f3</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X3</font>,<font color="Maroon">Y</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:151"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font> or <font color="Maroon">f1</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:151"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X3</font> or <font color="Maroon">f1</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:151"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X3</font> or <font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f3</font>;<br/>
<a NAME="E4:151"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T21">TSEP_1:21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f3</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span></span>)</span>,<font color="Maroon">Y</font> ;<br/>
<a NAME="E6:151"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<a NAME="E7:151"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<a NAME="E8:151"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<a NAME="E9:151"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:151_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="151_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:151_1_1_1"/>
( ( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; not <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> ) or ( not <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> ) or ( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> ) )
 </span><b>by </b><i><a class="ref" href="tmap_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tmap_1.html#D1" target="_self">Def1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="151_1_1_1_1"><b>suppose </b></a><a NAME="E1:151_1_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; not <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> )
 ;<br/><div class="add"><a NAME="E2:151_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font>
 <b>by </b><i/>;<br/><a NAME="E3:151_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/><a NAME="E4:151_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:151_1_1_1_1_1"/>
<font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E3:151_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T33" target="_self">Th33</a>, <a class="ref" href="tmap_1.html#T44" target="_self">Th44</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:151_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T46" target="_self">Th46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="151_1_1_1_2"><b>suppose </b></a><a NAME="E1:151_1_1_1_2"/><i><font color="Green">E48</font></i>: 
( not <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> )
 ;<br/><div class="add"><a NAME="E2:151_1_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font>
 <b>by </b><i/>;<br/><a NAME="E3:151_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/><a NAME="E4:151_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:151_1_1_1_2_1"/>
<font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E3:151_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T44" target="_self">Th44</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:151_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T46" target="_self">Th46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="151_1_1_1_3"><b>suppose </b></a><a NAME="E1:151_1_1_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> )
 ;<br/><div class="add"><a NAME="E2:151_1_1_1_3"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/><a NAME="E3:151_1_1_1_3"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/><a NAME="E4:151_1_1_1_3"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X3</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X3</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/><a NAME="E5:151_1_1_1_3"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X3</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:151"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E6:151_1_1_1_3"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:151_1_1_1_3_1"/>
<font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E3:151_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T33" target="_self">Th33</a>, <a class="ref" href="tmap_1.html#T44" target="_self">Th44</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:151_1_1_1_3"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:151_1_1_1_3_2"/>
<font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E3:151_1_1_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T44" target="_self">Th44</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">v</font> = <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span>;<br/><a NAME="E8:151_1_1_1_3"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/><a NAME="E9:151_1_1_1_3"/><b>then </b><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tmap_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T35">TSEP_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p4">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span> <a href="tmap_1.html#K10" target="_self">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p4">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T47" target="_self">Th47</a>, <a class="ref" href="tmap_1.html#T48" target="_self">Th48</a>, , <a class="ref" href="tmap_1.html#T29" target="_self">Th29</a></i>
<br/>.= 
<font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T29" target="_self">Th29</a></i>
;<br/><b>hence </b><a NAME="E11:151_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T35">TSEP_1:35</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:151_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X3</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E11:151"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f3</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f3</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X3</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f3</font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E12:151"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, <a class="ref" href="tmap_1.html#T38" target="_self">Th38</a>, </i>;<br/>
<a NAME="E13:151"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<a NAME="E14:151"/>
<font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>then </b><font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T37" target="_self">Th37</a>, <a class="ref" href="tmap_1.html#T43" target="_self">Th43</a>, </i>
;<br/>
<a NAME="E16:151"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E17:151"/>
<font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>then </b><i><font color="Green">E177</font></i>: <font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T37" target="_self">Th37</a>, <a class="ref" href="tmap_1.html#T43" target="_self">Th43</a>, </i>
;<br/>
<a NAME="E19:151"/>
<font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_4"><b>proof </b></a><div class="add">

<a NAME="E1:151_4"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>then </b><i><font color="Green">E179</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> = 
<font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">f2</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, , <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X3</font> = 
<font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X3</font>
<b>by </b><i><a class="ref" href="tmap_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">f3</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T39" target="_self">Th39</a>, <a class="ref" href="tmap_1.html#T43" target="_self">Th43</a>, </i>
;<br/>
<b>hence </b><a NAME="E4:151_4"/>
<font color="Maroon">f</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X3</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f3</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T47" target="_self">Th47</a>, <a class="ref" href="tmap_1.html#T29" target="_self">Th29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:151"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f3</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tmap_1.html#T29" target="_self">Th29</a></i>;<br/>


</div>
