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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:149"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a>, <a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>thus </b><a NAME="E3:149"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="149_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E107</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="149_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:149_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:149_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23">TSEP_1:23</a></i>;<br/><a NAME="E3:149_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:149"><i><font color="Green">E106</font></i></a>, <a class="ref" href="tmap_1.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E4:149_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E5:149_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:149_1_1"><i><font color="Green">E108</font></i></a></i>;<br/><a NAME="E6:149_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E7:149_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T74" target="_self">Th74</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="149_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:149_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><i><font color="Green">E109</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

;<br/> <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>

;<br/><a NAME="E4:149_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font> #)
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:149_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="tmap_1.html#E2:149_1_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="tsep_1.html#T5">TSEP_1:5</a></i>;<br/><b>hence </b><a NAME="E5:149_1_2"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23">TSEP_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:149_1"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:149_1"><i><font color="Green">E107</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E4:149"/>
( <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="149_2"><b>proof </b></a><div class="add">

<div><i><font color="Green">E107</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="149_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:149_2_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:149_2_1"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23">TSEP_1:23</a></i>;<br/><a NAME="E3:149_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:149"><i><font color="Green">E106</font></i></a>, <a class="ref" href="tmap_1.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E4:149_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E5:149_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:149_2_1"><i><font color="Green">E108</font></i></a></i>;<br/><a NAME="E6:149_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E7:149_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T74" target="_self">Th74</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="149_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:149_2_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><i><font color="Green">E109</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

;<br/> <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>

;<br/><a NAME="E4:149_2_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> #)
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:149_2_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="tmap_1.html#E2:149_2_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="tsep_1.html#T5">TSEP_1:5</a></i>;<br/><b>hence </b><a NAME="E5:149_2_2"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23">TSEP_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:149_2"/>
( <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:149_2"><i><font color="Green">E107</font></i></a></i>;<br/>


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


</div>
