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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font>, <font color="Olive" title="b2">V</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> <br/> for <font color="Olive" title="b3">B</font> being 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="Olive" title="b1">S</font><br/> for <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">B</font></span>)</span><br/> for <font color="Olive" title="b5">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">V</font><br/> for <font color="Olive" title="b6">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">V</font>  st <font color="Olive" title="b6">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">f</font> &amp; <font color="Olive" title="b4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b5">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> holds <br/><font color="Olive" title="b6">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/><b>let </b><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#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being 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/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</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="Olive" title="b1">B</font></span>)</span><br/> for <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b4">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b3">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> holds <br/><font color="Olive" title="b4">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/><b>let </b><font color="Maroon" title="c4">B</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</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">B</font></span>)</span><br/> for <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b3">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b2">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> holds <br/><font color="Olive" title="b3">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</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">B</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b2">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b2">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> holds <br/><font color="Olive" title="b2">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/><b>let </b><font color="Maroon" title="c6">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <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">X</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b1">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c6">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> holds <br/><font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/><b>let </b><font color="Maroon" title="c7">h</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c6">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> implies <font color="Maroon" title="c7">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c7">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c6">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c7">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><a NAME="E1:6_1"/><i><font color="Green" title="E8">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T181" title="RELAT_1:th.181">RELAT_1:181</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><a NAME="E2:6_1_1"/><b>then </b><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E2:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/><a NAME="E3:6_1_1"/><i><font color="Green" title="E10">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="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</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="E4:6_1_1"/><i><font color="Green" title="E11">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c4">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font></span>)</span> as   <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="c4">B</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E6:6_1_1"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c9">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E2:6_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="#E3:6_1_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></i>;<br/><a NAME="E7:6_1_1"/><i><font color="Green" title="E13">A8</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><font color="Maroon" title="c7">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> = 
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E1:6_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<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">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:6_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">F</font>
<b>by </b><i><a class="txt" href="#E4:6_1_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
;<br/><b>hence </b><a NAME="E9:6_1_1"/>
<font color="Maroon" title="c7">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E6:6_1_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:6_1"/>
( <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c7">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6"/>
<font color="Maroon" title="c7">h</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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