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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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="c5">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="c6">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>assume </b><a NAME="E1:145"/><i><font color="Green" title="E106">A1</font></i>: 
<font color="Maroon" title="c5">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="c6">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>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">Y1</font> = <font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c8">Y2</font> = <font color="Maroon" title="c4">X2</font>, <font color="Maroon" title="c9">Y3</font> = <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> as    <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#T2" title="TSEP_1:th.2">TSEP_1:2</a>, <a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/>
<b>thus </b><a NAME="E3:145"/>
( <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="c4">X2</font> iff <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E107">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_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="c4">X2</font>
 ;<br/><a NAME="E2:145_1_1"/><b>then </b><i><font color="Green" title="E108">A3</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">X2</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> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></i>;<br/><a NAME="E3:145_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145"><i><font color="Green" title="E106">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="E4:145_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <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="c8">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D4" target="_self" title="TMAP_1:def.4">Def4</a></i>;<br/><a NAME="E5:145_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <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="c9">Y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:145_1_1"><i><font color="Green" title="E108">A3</font></i></a></i>;<br/><a NAME="E6:145_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">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#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D4" target="_self" title="TMAP_1:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E7:145_1_1"/>
<font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T74" target="_self" title="TMAP_1:th.74">Th74</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_1_2"/><i><font color="Green" title="E108">A4</font></i>: 
<font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 ;<br/><a NAME="E2:145_1_2"/><i><font color="Green" title="E109">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:145_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:145_1_2"/><b>then </b>
<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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">X2</font> #)
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145_1_2"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="tmap_1.html#E2:145_1_2"><i><font color="Green" title="E109">A5</font></i></a>, <a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></i>;<br/><b>hence </b><a NAME="E5:145_1_2"/>
<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="c4">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:145_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="c4">X2</font> iff <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145_1"><i><font color="Green" title="E107">A2</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E4:145"/>
( <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> iff <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_2"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E107">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="145_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_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>
 ;<br/><a NAME="E2:145_2_1"/><b>then </b><i><font color="Green" title="E108">A7</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">X1</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> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></i>;<br/><a NAME="E3:145_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <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="c5">f1</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145"><i><font color="Green" title="E106">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="E4:145_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <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="c7">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D4" target="_self" title="TMAP_1:def.4">Def4</a></i>;<br/><a NAME="E5:145_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <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="c9">Y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:145_2_1"><i><font color="Green" title="E108">A7</font></i></a></i>;<br/><a NAME="E6:145_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">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#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D4" target="_self" title="TMAP_1:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E7:145_2_1"/>
<font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T74" target="_self" title="TMAP_1:th.74">Th74</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_2_2"/><i><font color="Green" title="E108">A8</font></i>: 
<font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font>
 ;<br/><a NAME="E2:145_2_2"/><i><font color="Green" title="E109">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:145_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:145_2_2"/><b>then </b>
<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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">X1</font> #)
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145_2_2"><i><font color="Green" title="E108">A8</font></i></a>, <a class="txt" href="tmap_1.html#E2:145_2_2"><i><font color="Green" title="E109">A9</font></i></a>, <a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></i>;<br/><b>hence </b><a NAME="E5:145_2_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>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:145_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> iff <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f1</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:145_2"><i><font color="Green" title="E107">A6</font></i></a></i>;<br/>


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


</div>
