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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c4">P1</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c5">P2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">P1</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">P2</font></span>)</span>,<font color="Maroon" title="c3">V</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c4">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P2</font>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c7">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">P3</font> = <font color="Maroon" title="c5">P2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="jordan6.html#E1:50"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c7">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P3</font></span>)</span>,<font color="Maroon" title="c3">V</font> ;<br/>
<a NAME="E6:50"/><i><font color="Green" title="E35">A4</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E7:50"/><i><font color="Green" title="E36">A5</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">P2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">f1</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P3</font></span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E1:50"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="jordan6.html#E6:50"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<a NAME="E9:50"/>
<font color="Maroon" title="c10">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1"/>
for <font color="Olive" title="b1">F1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">P2</font></span>)</span>  st <font color="Olive" title="b1">F1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> holds <br/><font color="Maroon" title="c10">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">F1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">F1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">P2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:50_1_1"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c11">F1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c12">P1'</font> = <font color="Maroon" title="c4">P1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P3</font></span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E1:50"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="jordan6.html#E7:50"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/>
<a NAME="E3:50_1_1"/><i><font color="Green" title="E38">A7</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">P4</font> = <font color="Maroon" title="c11">F1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">P1'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P3</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P1'</font></span>)</span> <b>by </b><i><a class="ref" href="tops_2.html#T38" title="TOPS_2:th.38">TOPS_2:38</a></i>;<br/>
<a NAME="E5:50_1_1"/><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P3</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P1'</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:50"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E6:50_1_1"/><i><font color="Green" title="E40">A9</font></i>: 
<font color="Maroon" title="c13">P4</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:50_1_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="jordan6.html#T3" target="_self" title="JORDAN6:th.3">Th3</a></i>;<br/>
<i><font color="Green" title="E41">A10</font></i>: <font color="Maroon" title="c10">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="txt" href="jordan6.html#E3:50_1_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="funct_2.html#T48" title="FUNCT_2:th.48">FUNCT_2:48</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f1</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="E8:50_1_1"/>
<font color="Maroon" title="c10">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a></i>;<br/>
<b>then </b><font color="Maroon" title="c10">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">F1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">F1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E7:50_1_1"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c13">P4</font>
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
;<br/>
<b>hence </b><a NAME="E10:50_1_1"/>
<font color="Maroon" title="c10">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">F1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E2:50"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="jordan6.html#E5:50_1_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="jordan6.html#E6:50_1_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1"/>
<font color="Maroon" title="c10">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:50"/><b>then </b>
<font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E3:50"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<b>hence </b><a NAME="E11:50"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">V</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 ;<br/>


</div>
