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<span class="kw">let </span><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">T</font>,<font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">S</font>  st <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">B</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <font color="Olive" title="b2">B</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <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="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Olive" title="b1">B</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <font color="Olive" title="b1">B</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c4">B</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  implies <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <font color="Maroon" title="c4">B</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:5"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E2:5"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c4">B</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <font color="Maroon" title="c4">B</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> </span><br/>

<a NAME="E3:5"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c5">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">thus </span><a NAME="E2:5_1"/>
<font color="Maroon" title="c5">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c6">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E2:5_1_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E3:5_1_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="funct_2.html#D10" title="FUNCT_2:def.10">FUNCT_2:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c6">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E5:5_1_1"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c7">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E2:5_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:5_1_1"/>
<font color="Maroon" title="c5">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:5_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:5"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <font color="Maroon" title="c4">B</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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