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<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">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">X0</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">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X0</font>,<font color="Maroon">Y</font>;<br/>







<a NAME="E1:64"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <font color="Olive">h</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_1"/>
(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</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">X0</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X0</font> #) or  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X0</font> #) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</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">X0</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X0</font> #)
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> ;<br/><b>take </b><font color="Maroon">h</font> = <font color="Maroon">h</font>;<br/><b>thus </b><a NAME="E3:64_1_1_1_1"/>
<font color="Maroon">h</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_1_2"/><i><font color="Green">E29</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X0</font> #)
 ;<br/><div class="add"><a NAME="E2:64_1_1_1_2"/>
<font color="Maroon">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">B</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/><a NAME="E4:64_1_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">A0</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><a NAME="E8:64_1_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A0</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E9:64_1_1_1_2"/>
<font color="Maroon">A0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="tsep_1.html#T5">TSEP_1:5</a></i>;<br/><a NAME="E10:64_1_1_1_2"/><b>then </b>
not <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A0</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">A0</font> = <font color="Maroon">A0</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font>;<br/><b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">A1</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">y</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A1</font>,<font color="Maroon">B</font> ;<br/><b>reconsider </b><font color="Maroon">g0</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A0</font>,<font color="Maroon">B</font> ;<br/><b>set </b><font color="Maroon">G</font> = <font color="Maroon">g0</font> <a href="tmap_1.html#K1" target="_self">union</a> <font color="Maroon">g1</font>;<br/><a NAME="E16:64_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g0</font> <a href="tmap_1.html#K1" target="_self">union</a> <font color="Maroon">g1</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g0</font>
 <b>by </b><i>, </i>;<br/><a NAME="E17:64_1_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A0</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">g0</font> <a href="tmap_1.html#K1" target="_self">union</a> <font color="Maroon">g1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> ;<br/><b>take </b><font color="Maroon">h</font> = <font color="Maroon">h</font>;<br/><b>thus </b><a NAME="E19:64_1_1_1_2"/>
<font color="Maroon">h</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:64_1"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <font color="Olive">h</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:64"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <font color="Olive">h</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 ;<br/>


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