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<a NAME="E1:145_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>set </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>;<br/>
<b>set </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>;<br/>
<b>set </b><font color="Maroon">A</font> = 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>;<br/>
<b>set </b><font color="Maroon">B</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>;<br/>
<a NAME="E2:145_1_1"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<a NAME="E3:145_1_1"/><i><font color="Green">E36</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> implies <font color="Maroon">f1</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:145_1_1_1"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E2:145_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>
<a NAME="E3:145_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/>
<b>thus </b><font color="Maroon">f1</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font></span>)</span> = 
<font color="Maroon">f1</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tsep_1.html#D5">TSEP_1:def 5</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T33" target="_self">Th33</a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, , <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, </i>
<br/>.= 
<font color="Maroon">f2</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tsep_1.html#D5">TSEP_1:def 5</a></i>
;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>, <font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of 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="tmap_1.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">f1</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">f2</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> ;<br/>
<b>set </b><font color="Maroon">G</font> = <font color="Maroon">g1</font> <a href="tmap_1.html#K1" target="_self">union</a> <font color="Maroon">g2</font>;<br/>
<a NAME="E7:145_1_1"/><i><font color="Green">E43</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="tmap_1.html#K1" target="_self">union</a> <font color="Maroon">g2</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="tmap_1.html#K1" target="_self">union</a> <font color="Maroon">g2</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, , </i>;<br/>
<a NAME="E8:145_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">g1</font> <a href="tmap_1.html#K1" target="_self">union</a> <font color="Maroon">g2</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">Y</font> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E10:145_1_1"/>
( <font color="Maroon">F</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">F</font> <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#T29" target="_self">Th29</a>, <a class="ref" href="tmap_1.html#T33" target="_self">Th33</a>, </i>;<br/>


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