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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">X2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Maroon">Y</font>;<br/>









<b>assume </b><a NAME="E1:149"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f1</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">Y1</font> = <font color="Maroon">X1</font>, <font color="Maroon">Y2</font> = <font color="Maroon">X2</font>, <font color="Maroon">Y3</font> = <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> as    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> <b>by </b><i><a class="ref" href="tsep_1.html#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">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> iff <font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f2</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">E29</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">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font>
 ;<br/><a NAME="E2:149_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X2</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23">TSEP_1:23</a></i>;<br/><a NAME="E3:149_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E4:149_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i/>;<br/><a NAME="E5:149_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E6:149_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:149_1_1"/>
<font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><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">E36</font></i>: 
<font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f2</font>
 ;<br/><a NAME="E2:149_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:149_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:149_1_2"/><b>then </b>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</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">X2</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X2</font> #)
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</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">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23">TSEP_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:149_1"/>
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> iff <font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T33" target="_self">Th33</a></i>;<br/>


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

<b>thus </b><a NAME="E4:149"/>
( <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> iff <font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</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">E39</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">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font>
 ;<br/><a NAME="E2:149_2_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23">TSEP_1:23</a></i>;<br/><a NAME="E3:149_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E4:149_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font>
 <b>by </b><i/>;<br/><a NAME="E5:149_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E6:149_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:149_2_1"/>
<font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font>
 <b>by </b><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">E43</font></i>: 
<font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font>
 ;<br/><a NAME="E2:149_2_2"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:149_2_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:149_2_2"/><b>then </b>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</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">X1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font> #)
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</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">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</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">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> iff <font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T33" target="_self">Th33</a></i>;<br/>


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


</div>
