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<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>, <font color="Maroon" title="c5">X3</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>let </b><font color="Maroon" title="c6">f1</font> be   <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>;<br/><b>let </b><font color="Maroon" title="c7">f2</font> be   <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/><b>let </b><font color="Maroon" title="c8">f3</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">X3</font>,<font color="Maroon" title="c2">Y</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:145"/><i><font color="Green" title="E96">A1</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> or <font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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>and </b><br/><a NAME="E2:145"/><i><font color="Green" title="E97">A2</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="c5">X3</font> or <font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:145"/><i><font color="Green" title="E98">A3</font></i>: 
( <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c5">X3</font> or <font color="Maroon" title="c7">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon" title="c9">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c8">f3</font>;<br/>
<a NAME="E4:145"/><i><font color="Green" title="E99">A4</font></i>: 
<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="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T21" title="TSEP_1:th.21">TSEP_1:21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c8">f3</font> as   <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">Y</font> ;<br/>
<a NAME="E6:145"/><i><font color="Green" title="E100">A5</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 <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="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</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="E7:145"/><i><font color="Green" title="E101">A6</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="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:145"><i><font color="Green" title="E99">A4</font></i></a>, <a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/>
<a NAME="E8:145"/><i><font color="Green" title="E102">A7</font></i>: 
<font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</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="E9:145"/><i><font color="Green" title="E103">A8</font></i>: 
<font color="Maroon" title="c5">X3</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:145"><i><font color="Green" title="E99">A4</font></i></a>, <a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_1"/><i><font color="Green" title="E104">A9</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> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="145_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:145_1_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="c5">X3</font> &amp; not <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font> ) or ( not <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font> ) or ( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font> ) )
 </span><b>by </b><i><a class="txt" href="tmap_1.html#E1:145_1"><i><font color="Green" title="E104">A9</font></i></a>, <a class="ref" href="tmap_1.html#T39" target="_self" title="TMAP_1:th.39">Th39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="145_1_1_1_1"><b>suppose </b></a><a NAME="E1:145_1_1_1_1"/><i><font color="Green" title="E105">A10</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="c5">X3</font> &amp; not <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font> )
 ;<br/><div class="add"><a NAME="E2:145_1_1_1_1"/><b>then </b><i><font color="Green" title="E106">A11</font></i>: 
<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="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T31" target="_self" title="TMAP_1:th.31">Th31</a></i>;<br/><a NAME="E3:145_1_1_1_1"/><i><font color="Green" title="E107">A12</font></i>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145_1_1_1_1"><i><font color="Green" title="E105">A10</font></i></a>, <a class="ref" href="tsep_1.html#T30" title="TSEP_1:th.30">TSEP_1:30</a></i>;<br/><a NAME="E4:145_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:145_1_1_1_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="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>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E3:145_1_1_1_1"><i><font color="Green" title="E107">A12</font></i></a>, <a class="ref" href="tmap_1.html#T79" target="_self" title="TMAP_1:th.79">Th79</a></i>
<br/>.= 
<font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E1:145"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="tmap_1.html#D12" target="_self" title="TMAP_1:def.12">Def12</a></i>
;<br/>
<b>hence </b><a NAME="E3:145_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:145"><i><font color="Green" title="E97">A2</font></i></a>, <a class="txt" href="tmap_1.html#E1:145_1_1_1_1"><i><font color="Green" title="E105">A10</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:145_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:145_1_1_1_1"><i><font color="Green" title="E106">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="145_1_1_1_2"><b>suppose </b></a><a NAME="E1:145_1_1_1_2"/><i><font color="Green" title="E105">A13</font></i>: 
( not <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font> )
 ;<br/><div class="add"><a NAME="E2:145_1_1_1_2"/><b>then </b><i><font color="Green" title="E106">A14</font></i>: 
<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="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T31" target="_self" title="TMAP_1:th.31">Th31</a></i>;<br/><a NAME="E3:145_1_1_1_2"/><i><font color="Green" title="E107">A15</font></i>: 
<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145_1_1_1_2"><i><font color="Green" title="E105">A13</font></i></a>, <a class="ref" href="tsep_1.html#T30" title="TSEP_1:th.30">TSEP_1:30</a></i>;<br/><a NAME="E4:145_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:145_1_1_1_2_1"/>
<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>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E3:145_1_1_1_2"><i><font color="Green" title="E107">A15</font></i></a>, <a class="ref" href="tmap_1.html#T79" target="_self" title="TMAP_1:th.79">Th79</a></i>
<br/>.= 
<font color="Maroon" title="c7">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E1:145"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="tmap_1.html#D12" target="_self" title="TMAP_1:def.12">Def12</a></i>
;<br/>
<b>hence </b><a NAME="E3:145_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E3:145"><i><font color="Green" title="E98">A3</font></i></a>, <a class="txt" href="tmap_1.html#E1:145_1_1_1_2"><i><font color="Green" title="E105">A13</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:145_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:145_1_1_1_2"><i><font color="Green" title="E106">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="145_1_1_1_3"><b>suppose </b></a><a NAME="E1:145_1_1_1_3"/><i><font color="Green" title="E105">A16</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="c5">X3</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c5">X3</font> )
 ;<br/><div class="add"><a NAME="E2:145_1_1_1_3"/><b>then </b><i><font color="Green" title="E106">A17</font></i>: 
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T30" title="TSEP_1:th.30">TSEP_1:30</a></i>;<br/><a NAME="E3:145_1_1_1_3"/><b>then </b><i><font color="Green" title="E107">A18</font></i>: 
<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="c5">X3</font></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="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</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#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#T27" target="_self" title="TMAP_1:th.27">Th27</a></i>;<br/><a NAME="E4:145_1_1_1_3"/>
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c5">X3</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c5">X3</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145_1_1_1_3"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="tsep_1.html#T30" title="TSEP_1:th.30">TSEP_1:30</a></i>;<br/><a NAME="E5:145_1_1_1_3"/><b>then </b><i><font color="Green" title="E108">A19</font></i>: 
<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="c5">X3</font></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="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c5">X3</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T29" target="_self" title="TMAP_1:th.29">Th29</a></i>;<br/><a NAME="E6:145_1_1_1_3"/><i><font color="Green" title="E109">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:145_1_1_1_3_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="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>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E2:145_1_1_1_3"><i><font color="Green" title="E106">A17</font></i></a>, <a class="ref" href="tmap_1.html#T79" target="_self" title="TMAP_1:th.79">Th79</a></i>
<br/>.= 
<font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E1:145"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="tmap_1.html#D12" target="_self" title="TMAP_1:def.12">Def12</a></i>
;<br/>
<b>hence </b><a NAME="E3:145_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:145"><i><font color="Green" title="E97">A2</font></i></a>, <a class="txt" href="tmap_1.html#E1:145_1_1_1_3"><i><font color="Green" title="E105">A16</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:145_1_1_1_3"/><i><font color="Green" title="E110">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:145_1_1_1_3_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> <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>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E2:145_1_1_1_3"><i><font color="Green" title="E106">A17</font></i></a>, <a class="ref" href="tmap_1.html#T79" target="_self" title="TMAP_1:th.79">Th79</a></i>
<br/>.= 
<font color="Maroon" title="c7">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E1:145"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="tmap_1.html#D12" target="_self" title="TMAP_1:def.12">Def12</a></i>
;<br/>
<b>hence </b><a NAME="E3:145_1_1_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E3:145"><i><font color="Green" title="E98">A3</font></i></a>, <a class="txt" href="tmap_1.html#E1:145_1_1_1_3"><i><font color="Green" title="E105">A16</font></i></a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><span class="p2">(<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="c5">X3</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span>;<br/><a NAME="E8:145_1_1_1_3"/>
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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="c5">X3</font></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="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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="c5">X3</font></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="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> )
 <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="E9:145_1_1_1_3"/><b>then </b><i><font color="Green" title="E111">A22</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c5">X3</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c5">X3</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c5">X3</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <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="c5">X3</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p2">(<span class="default"><span class="p3">(<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="c5">X3</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E3:145_1_1_1_3"><i><font color="Green" title="E107">A18</font></i></a>, <a class="txt" href="tmap_1.html#E5:145_1_1_1_3"><i><font color="Green" title="E108">A19</font></i></a>, <a class="ref" href="tmap_1.html#T79" target="_self" title="TMAP_1:th.79">Th79</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><span class="p2">(<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="c5">X3</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E1:145_1_1_1_3"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="tsep_1.html#T35" title="TSEP_1:th.35">TSEP_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c5">X3</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p2">(<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="c5">X3</font></span>)</span></span>)</span> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p3">(<span class="default"><span class="p4">(<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="c5">X3</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E6:145_1_1_1_3"><i><font color="Green" title="E109">A20</font></i></a>, <a class="txt" href="tmap_1.html#E7:145_1_1_1_3"><i><font color="Green" title="E110">A21</font></i></a>, <a class="txt" href="tmap_1.html#E9:145_1_1_1_3"><i><font color="Green" title="E111">A22</font></i></a>, <a class="ref" href="tmap_1.html#T138" target="_self" title="TMAP_1:th.138">Th138</a></i>
<br/>.= 
<font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><span class="p2">(<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="c5">X3</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T138" target="_self" title="TMAP_1:th.138">Th138</a></i>
;<br/><b>hence </b><a NAME="E11:145_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145_1_1_1_3"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="tsep_1.html#T35" title="TSEP_1:th.35">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="E3:145_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E11:145"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c8">f3</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c8">f3</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c5">X3</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f3</font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#D12" target="_self" title="TMAP_1:def.12">Def12</a></i>;<br/>
<a NAME="E12:145"/><b>then </b><i><font color="Green" title="E104">A23</font></i>: 
( <font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font> &amp; <font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f3</font> )
 
<b>by </b><i><a class="txt" href="tmap_1.html#E6:145"><i><font color="Green" title="E100">A5</font></i></a>, <a class="txt" href="tmap_1.html#E8:145"><i><font color="Green" title="E102">A7</font></i></a>, <a class="ref" href="tmap_1.html#D4" target="_self" title="TMAP_1:def.4">Def4</a></i>;<br/>
<a NAME="E13:145"/><i><font color="Green" title="E105">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="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:145"><i><font color="Green" title="E99">A4</font></i></a>, <a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/>
<a NAME="E14:145"/>
<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>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/>
<b>then </b><font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</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#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font>
<b>by </b><i><a class="txt" href="tmap_1.html#E13:145"><i><font color="Green" title="E105">A24</font></i></a>, <a class="ref" href="tmap_1.html#T79" target="_self" title="TMAP_1:th.79">Th79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font>
<b>by </b><i><a class="txt" href="tmap_1.html#E7:145"><i><font color="Green" title="E101">A6</font></i></a>, <a class="txt" href="tmap_1.html#E12:145"><i><font color="Green" title="E104">A23</font></i></a>, <a class="ref" href="tmap_1.html#D4" target="_self" title="TMAP_1:def.4">Def4</a></i>
;<br/>
<a NAME="E16:145"/><b>then </b><i><font color="Green" title="E106">A25</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f1</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:145"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="tmap_1.html#D12" target="_self" title="TMAP_1:def.12">Def12</a></i>;<br/>
<a NAME="E17:145"/>
<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>then </b><i><font color="Green" title="E107">A26</font></i>: <font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</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#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font>
<b>by </b><i><a class="txt" href="tmap_1.html#E13:145"><i><font color="Green" title="E105">A24</font></i></a>, <a class="ref" href="tmap_1.html#T79" target="_self" title="TMAP_1:th.79">Th79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font>
<b>by </b><i><a class="txt" href="tmap_1.html#E7:145"><i><font color="Green" title="E101">A6</font></i></a>, <a class="txt" href="tmap_1.html#E12:145"><i><font color="Green" title="E104">A23</font></i></a>, <a class="ref" href="tmap_1.html#D4" target="_self" title="TMAP_1:def.4">Def4</a></i>
;<br/>
<a NAME="E19:145"/>
<font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">f2</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c8">f3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_4"><b>proof </b></a><div class="add">

<a NAME="E1:145_4"/><i><font color="Green" title="E108">A27</font></i>: 
( <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="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font> &amp; <font color="Maroon" title="c5">X3</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/>
<b>then </b><i><font color="Green" title="E109">A28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> = 
<font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T79" target="_self" title="TMAP_1:th.79">Th79</a></i>
<br/>.= 
<font color="Maroon" title="c7">f2</font>
<b>by </b><i><a class="txt" href="tmap_1.html#E1:145"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="tmap_1.html#E18:145"><i><font color="Green" title="E107">A26</font></i></a>, <a class="ref" href="tmap_1.html#D12" target="_self" title="TMAP_1:def.12">Def12</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c5">X3</font> = 
<font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c5">X3</font>
<b>by </b><i><a class="txt" href="tmap_1.html#E1:145_4"><i><font color="Green" title="E108">A27</font></i></a>, <a class="ref" href="tmap_1.html#T79" target="_self" title="TMAP_1:th.79">Th79</a></i>
<br/>.= 
<font color="Maroon" title="c8">f3</font>
<b>by </b><i><a class="txt" href="tmap_1.html#E9:145"><i><font color="Green" title="E103">A8</font></i></a>, <a class="txt" href="tmap_1.html#E12:145"><i><font color="Green" title="E104">A23</font></i></a>, <a class="ref" href="tmap_1.html#D4" target="_self" title="TMAP_1:def.4">Def4</a></i>
;<br/>
<b>hence </b><a NAME="E4:145_4"/>
<font color="Maroon" title="c10">f</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">X3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">f2</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c8">f3</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:145_4"><i><font color="Green" title="E109">A28</font></i></a>, <a class="ref" href="tmap_1.html#T138" target="_self" title="TMAP_1:th.138">Th138</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:145"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c8">f3</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c8">f3</font></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E16:145"><i><font color="Green" title="E106">A25</font></i></a>, <a class="ref" href="tmap_1.html#T138" target="_self" title="TMAP_1:th.138">Th138</a></i>;<br/>


</div>
