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<span class="kw">let </span><font color="Maroon" title="c1">T</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">ET</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">ET</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<a NAME="E1:33"/>
 ex <font color="Olive" title="b1">ET</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fintopo2.html#V1" title="FINTOPO2:attr.1">strict</a>   <a href="fintopo2.html#L1" title="FINTOPO2:struct.1">FMT_Space_Str</a>  st <br/>( <font color="Olive" title="b1">ET</font> is  <a href="fintopo7.html#V2" title="FINTOPO7:attr.2">U_FMT_filter</a>  &amp; <font color="Olive" title="b1">ET</font> is  <a href="fintopo7.html#V3" title="FINTOPO7:attr.3">U_FMT_with_point</a>  &amp; <font color="Olive" title="b1">ET</font> is  <a href="fintopo7.html#V4" title="FINTOPO7:attr.4">U_FMT_local</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">ET</font> &amp;  ex <font color="Olive" title="b2">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b2">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b2">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> : $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> </span> } </span> ;<br/>
<a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a></span>;<br/>
<a NAME="E3:33_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">x</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">x</font>) or <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_1_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">x</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">O1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E3:33_1_1_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">O1</font>
 <span class="kw">and </span><br/><a NAME="E4:33_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">O1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:33_1_1_1"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:33_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:33_1"/>
 ex <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> st <br/> for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>)
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" onmouseover="rs('funct_2/S2')" onmouseout="rh()">FUNCT_2:sch 2</a>(<a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E21">A1</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:33_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">TMP</font> =  <a href="fintopo2.html#G1" title="FINTOPO2:aggr.1">FMT_Space_Str</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">f</font> #) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fintopo2.html#V1" title="FINTOPO2:attr.1">strict</a>   <a href="fintopo2.html#L1" title="FINTOPO2:struct.1">FMT_Space_Str</a>  ;<br/>
<a NAME="E6:33_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 for <font color="Olive" title="b1">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">TMP</font>  holds  not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">t</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">TMP</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">t</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<span class="kw">take </span><font color="Maroon" title="c5">TT</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">TT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">t</font></span><br/>

<a NAME="E1:33_1_2"/>
<font color="Maroon" title="c5">TT</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D1" onmouseover="rs('pre_topc/D1')" onmouseout="rh()">PRE_TOPC:def 1</a></span>;<br/>
<a NAME="E2:33_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">TT</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">t</font>)
 
;<br/>
<span class="kw">hence </span><a NAME="E3:33_1_2"/>
<font color="Maroon" title="c5">TT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:33_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c3">TMP</font> is  <a href="fintopo7.html#V5" title="FINTOPO7:attr.5">U_FMT_filter_base</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_3"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TMP</font> holds <br/> (  not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; (  for <font color="Olive" title="b2">B1</font>, <font color="Olive" title="b3">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font>  ex <font color="Olive" title="b4">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> st <font color="Olive" title="b4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">B2</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TMP</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; (  for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font>  ex <font color="Olive" title="b3">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> st <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">B2</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:33_1_3_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; (  for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font>  ex <font color="Olive" title="b3">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> st <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">B2</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:33_1_3_1"/>
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font>  ex <font color="Olive" title="b3">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> st <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">B2</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:33_1_3_1_1"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
 not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:33_1_3_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_3_1_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">O</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:33_1_3_1_1"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<font color="Maroon" title="c5">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E5:33_1_3_1_1"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 ;<br/>
<span class="kw">thus </span><a NAME="E6:33_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1_3_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E5:33_1_3_1_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E3:33_1_3_1"/>
 for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font>  ex <font color="Olive" title="b3">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> st <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">B2</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">B1</font>, <font color="Maroon" title="c6">B2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B2</font></span><br/>

<a NAME="E1:33_1_3_1_2"/>
( <font color="Maroon" title="c5">B1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">B2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_3_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a></span>;<br/>
<a NAME="E2:33_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A12</font></span>: 
( <font color="Maroon" title="c5">B1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">x</font>) &amp; <font color="Maroon" title="c6">B2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">x</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">O1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:33_1_3_1_2"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">O1</font>
 <span class="kw">and </span><br/><a NAME="E5:33_1_3_1_2"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">O1</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">O2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E7:33_1_3_1_2"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<font color="Maroon" title="c6">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">O2</font>
 <span class="kw">and </span><br/><a NAME="E8:33_1_3_1_2"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">O2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_3_1_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a></span>;<br/>
<a NAME="E9:33_1_3_1_2"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">O1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_3_1_2"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E8:33_1_3_1_2"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">OO</font> = <font color="Maroon" title="c7">O1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">O2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D1" onmouseover="rs('pre_topc/D1')" onmouseout="rh()">PRE_TOPC:def 1</a></span>;<br/>
<a NAME="E11:33_1_3_1_2"/>
<font color="Maroon" title="c9">OO</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33_1_3_1_2"><span class="lab"><font color="Green" title="E31">A17</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">OO</font> = <font color="Maroon" title="c9">OO</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">x</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">OO</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">OO</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B2</font></span><br/>

<span class="kw">thus </span><a NAME="E13:33_1_3_1_2"/>
<font color="Maroon" title="c10">OO</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1_3_1_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E7:33_1_3_1_2"><span class="lab"><font color="Green" title="E29">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div><span class="kw">end;</span></div>
<a NAME="E2:33_1_3"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TMP</font> holds <br/> (  not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> is  <a href="cardfil2.html#V4" title="CARDFIL2:attr.4">quasi_basis</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:33_1_3"/>
<font color="Maroon" title="c3">TMP</font> is  <a href="fintopo7.html#V5" title="FINTOPO7:attr.5">U_FMT_filter_base</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c4">ET</font> =  <a href="fintopo7.html#K3" title="FINTOPO7:func.3">gen_filter</a> <font color="Maroon" title="c3">TMP</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fintopo2.html#V1" title="FINTOPO2:attr.1">strict</a>   <a href="fintopo2.html#L1" title="FINTOPO2:struct.1">FMT_Space_Str</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">ET</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V2" title="FINTOPO7:attr.2">U_FMT_filter</a>  &amp; <font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V3" title="FINTOPO7:attr.3">U_FMT_with_point</a>  &amp; <font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V4" title="FINTOPO7:attr.4">U_FMT_local</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">ET</font> &amp;  ex <font color="Olive" title="b1">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:33_1"/>
<font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V2" title="FINTOPO7:attr.2">U_FMT_filter</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="fintopo7.html#T13" target="_self" onmouseover="rs('fintopo7/T13')" onmouseout="rh()">Th8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V3" title="FINTOPO7:attr.3">U_FMT_with_point</a>  &amp; <font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V4" title="FINTOPO7:attr.4">U_FMT_local</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">ET</font> &amp;  ex <font color="Olive" title="b1">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:33_1"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
<font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V3" title="FINTOPO7:attr.3">U_FMT_with_point</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V4" title="FINTOPO7:attr.4">U_FMT_local</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">ET</font> &amp;  ex <font color="Olive" title="b1">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_4"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font><br/>  for <font color="Olive" title="b2">V</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> holds  <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">V</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">x</font> holds  <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">V</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">x</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">V</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c7">Z</font> =  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <span class="p1">(<span class="default"><a href="fintopo7.html#K3" title="FINTOPO7:func.3">gen_filter</a> <font color="Maroon" title="c3">TMP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x0</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E2:33_1_4_1"/><span class="lab"><font color="Green" title="E26">A20</font></span>: 
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c8">x0</font></span>)</span></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D9" target="_self" onmouseover="rs('fintopo7/D9')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E3:33_1_4_1"/><span class="lab"><font color="Green" title="E27">A21</font></span>: 
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c8">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">FX0</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">x0</font>) as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E5:33_1_4_1"/><span class="lab"><font color="Green" title="E28">A22</font></span>: 
<font color="Maroon" title="c6">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c9">FX0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E2:33_1_4_1"><span class="lab"><font color="Green" title="E26">A20</font></span></a></span>;<br/>
<a NAME="E6:33_1_4_1"/>
 not <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c9">FX0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_4_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D1" onmouseover="rs('pre_topc/D1')" onmouseout="rh()">PRE_TOPC:def 1</a></span>;<br/>
<a NAME="E2:33_1_4_1_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">FX0</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">XX</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">FX0</font> ;<br/>
<a NAME="E4:33_1_4_1_1"/>
<font color="Maroon" title="c9">FX0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c9">FX0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cardfil2.html#D8" onmouseover="rs('cardfil2/D8')" onmouseout="rh()">CARDFIL2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:33_1_4_1_1"/>
 not <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c9">FX0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_4_1"><span class="lab"><font color="Green" title="E27">A21</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:33_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c9">FX0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_4_1"><span class="lab"><font color="Green" title="E28">A22</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">FX0</font><span class="kw"> such that </span><br/><a NAME="E9:33_1_4_1"/><span class="lab"><font color="Green" title="E29">A23</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cardfil2.html#D8" onmouseover="rs('cardfil2/D8')" onmouseout="rh()">CARDFIL2:def 8</a></span>;<br/>
<a NAME="E10:33_1_4_1"/>
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">x0</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_4_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_4_1_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D1" onmouseover="rs('pre_topc/D1')" onmouseout="rh()">PRE_TOPC:def 1</a></span>;<br/>
<a NAME="E2:33_1_4_1_2"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">x0</font>)
 
;<br/>
<span class="kw">hence </span><a NAME="E3:33_1_4_1_2"/>
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">x0</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:33_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">x0</font>)
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">OO</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E13:33_1_4_1"/><span class="lab"><font color="Green" title="E30">A24</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">OO</font>
 <span class="kw">and </span><br/><a NAME="E14:33_1_4_1"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
<font color="Maroon" title="c8">x0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">OO</font>
 ;<br/>
<span class="kw">thus </span><a NAME="E15:33_1_4_1"/>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33_1_4_1"><span class="lab"><font color="Green" title="E29">A23</font></span></a>, <a class="txt" href="#E13:33_1_4_1"><span class="lab"><font color="Green" title="E30">A24</font></span></a>, <a class="txt" href="#E14:33_1_4_1"><span class="lab"><font color="Green" title="E31">A25</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:33_1_4"/>
<font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V3" title="FINTOPO7:attr.3">U_FMT_with_point</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E11:33_1"/><span class="lab"><font color="Green" title="E26">A26</font></span>: 
<font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V4" title="FINTOPO7:attr.4">U_FMT_local</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">ET</font> &amp;  ex <font color="Olive" title="b1">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_5"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font><br/>  for <font color="Olive" title="b2">V</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font>  ex <font color="Olive" title="b3">W</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> st <br/> for <font color="Olive" title="b4">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>  st <font color="Olive" title="b4">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b3">W</font> holds <br/><font color="Olive" title="b2">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b4">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font>  ex <font color="Olive" title="b2">W</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font> st <br/> for <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>  st <font color="Olive" title="b3">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">W</font> holds <br/><font color="Olive" title="b1">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b3">y</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">Z</font> =  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <span class="p1">(<span class="default"><a href="fintopo7.html#K3" title="FINTOPO7:func.3">gen_filter</a> <font color="Maroon" title="c3">TMP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">t</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">t0</font> = <font color="Maroon" title="c5">t</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E2:33_1_5_1"/><span class="lab"><font color="Green" title="E27">A28</font></span>: 
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c7">t0</font></span>)</span></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D9" target="_self" onmouseover="rs('fintopo7/D9')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E3:33_1_5_1"/><span class="lab"><font color="Green" title="E28">A29</font></span>: 
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c7">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">t0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">FT0</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">t0</font>) as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E5:33_1_5_1"/>
 for <font color="Olive" title="b1">V</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font>  ex <font color="Olive" title="b2">W</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font> st <br/> for <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>  st <font color="Olive" title="b3">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">W</font> holds <br/><font color="Olive" title="b1">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b3">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">V</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font> st <br/> for <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>  st <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">W</font> holds <br/><font color="Maroon" title="c9">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b2">y</font></span><br/>

<a NAME="E1:33_1_5_1_1"/><span class="lab"><font color="Green" title="E29">A30</font></span>: 
 not <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">FT0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_5_1_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D1" onmouseover="rs('pre_topc/D1')" onmouseout="rh()">PRE_TOPC:def 1</a></span>;<br/>
<a NAME="E2:33_1_5_1_1_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">FT0</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">XX</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">FT0</font> ;<br/>
<a NAME="E4:33_1_5_1_1_1"/>
<font color="Maroon" title="c10">XX</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">FT0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_5_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_5_1_1_1_1"/>
( <font color="Maroon" title="c10">XX</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">FT0</font> &amp; <font color="Maroon" title="c10">XX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">XX</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E2:33_1_5_1_1_1_1"/>
<font color="Maroon" title="c10">XX</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">FT0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cardfil2.html#D8" onmouseover="rs('cardfil2/D8')" onmouseout="rh()">CARDFIL2:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:33_1_5_1_1_1"/>
 not <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">FT0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:33_1_5_1_1"/><span class="lab"><font color="Green" title="E30">A31</font></span>: 
<font color="Maroon" title="c9">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">FT0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_5_1_1"><span class="lab"><font color="Green" title="E29">A30</font></span></a>, <a class="txt" href="#E2:33_1_5_1"><span class="lab"><font color="Green" title="E27">A28</font></span></a>, <a class="txt" href="#E3:33_1_5_1"><span class="lab"><font color="Green" title="E28">A29</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">V0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">FT0</font><span class="kw"> such that </span><br/><a NAME="E4:33_1_5_1_1"/><span class="lab"><font color="Green" title="E31">A32</font></span>: 
<font color="Maroon" title="c10">V0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_5_1_1"><span class="lab"><font color="Green" title="E30">A31</font></span></a>, <a class="ref" href="cardfil2.html#D8" onmouseover="rs('cardfil2/D8')" onmouseout="rh()">CARDFIL2:def 8</a></span>;<br/>
<a NAME="E5:33_1_5_1_1"/><span class="lab"><font color="Green" title="E32">A33</font></span>: 
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">t0</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_5_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_5_1_1_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D1" onmouseover="rs('pre_topc/D1')" onmouseout="rh()">PRE_TOPC:def 1</a></span>;<br/>
<a NAME="E2:33_1_5_1_1_2"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">t0</font>)
 
;<br/>
<span class="kw">hence </span><a NAME="E3:33_1_5_1_1_2"/>
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">t0</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:33_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">V0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">t0</font>)
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">OUV</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E8:33_1_5_1_1"/><span class="lab"><font color="Green" title="E33">A34</font></span>: 
<font color="Maroon" title="c10">V0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">OUV</font>
 <span class="kw">and </span><br/><a NAME="E9:33_1_5_1_1"/><span class="lab"><font color="Green" title="E34">A35</font></span>: 
<font color="Maroon" title="c7">t0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">OUV</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">W</font> = <font color="Maroon" title="c11">OUV</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_5_1"><span class="lab"><font color="Green" title="E27">A28</font></span></a>, <a class="txt" href="#E3:33_1_5_1"><span class="lab"><font color="Green" title="E28">A29</font></span></a>, <a class="txt" href="#E8:33_1_5_1_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="ref" href="cardfil2.html#D8" onmouseover="rs('cardfil2/D8')" onmouseout="rh()">CARDFIL2:def 8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">W</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>  st <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">W</font> holds <br/><font color="Maroon" title="c9">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">y</font></span><br/>

<a NAME="E11:33_1_5_1_1"/>
 for <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>  st <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">W</font> holds <br/><font color="Maroon" title="c9">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_5_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">W</font> implies <font color="Maroon" title="c9">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c13">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_5_1_1_3"/><span class="lab"><font color="Green" title="E35">A36</font></span>: 
<font color="Maroon" title="c13">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">W</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c13">y</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c14">Z</font> =  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <span class="p1">(<span class="default"><a href="fintopo7.html#K3" title="FINTOPO7:func.3">gen_filter</a> <font color="Maroon" title="c3">TMP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">y0</font> = <font color="Maroon" title="c13">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E3:33_1_5_1_1_3"/><span class="lab"><font color="Green" title="E36">A38</font></span>: 
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c15">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">y0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">FY0</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">y0</font>) as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E5:33_1_5_1_1_3"/><span class="lab"><font color="Green" title="E37">A39</font></span>: 
<font color="Maroon" title="c10">V0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">y0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1_5_1_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E1:33_1_5_1_1_3"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E9:33_1_5_1_1"><span class="lab"><font color="Green" title="E34">A35</font></span></a></span>;<br/>
<a NAME="E6:33_1_5_1_1_3"/>
( <font color="Maroon" title="c10">V0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">FT0</font> &amp; <font color="Maroon" title="c8">FT0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TMP</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_5_1_1"><span class="lab"><font color="Green" title="E32">A33</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">VV0</font> = <font color="Maroon" title="c10">V0</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E8:33_1_5_1_1_3"/>
<font color="Maroon" title="c9">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c16">FY0</font></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_5_1_1_3"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="txt" href="#E4:33_1_5_1_1"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="txt" href="#E2:33_1_5_1_1"><span class="lab"><font color="Green" title="E30">A31</font></span></a>, <a class="ref" href="cardfil2.html#D8" onmouseover="rs('cardfil2/D8')" onmouseout="rh()">CARDFIL2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:33_1_5_1_1_3"/>
<font color="Maroon" title="c9">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D9" target="_self" onmouseover="rs('fintopo7/D9')" onmouseout="rh()">Def7</a>, <a class="txt" href="#E3:33_1_5_1_1_3"><span class="lab"><font color="Green" title="E36">A38</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:33_1_5_1_1"/>
 for <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>  st <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">W</font> holds <br/><font color="Maroon" title="c9">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:33_1_5_1"/>
 for <font color="Olive" title="b1">V</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font>  ex <font color="Olive" title="b2">W</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">t</font> st <br/> for <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>  st <font color="Olive" title="b3">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">W</font> holds <br/><font color="Olive" title="b1">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b3">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:33_1_5"/>
<font color="Maroon" title="c4">ET</font> is  <a href="fintopo7.html#V4" title="FINTOPO7:attr.4">U_FMT_local</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E12:33_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">ET</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<a NAME="E13:33_1"/>
 ex <font color="Olive" title="b1">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">TT</font> = <font color="Maroon" title="c4">ET</font> as   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="fintopo7.html#T13" target="_self" onmouseover="rs('fintopo7/T13')" onmouseout="rh()">Th8</a>, <a class="txt" href="#E10:33_1"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="txt" href="#E11:33_1"><span class="lab"><font color="Green" title="E26">A26</font></span></a></span>;<br/>
<a NAME="E2:33_1_6"/>
 <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_6_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_6_1"/><span class="lab"><font color="Green" title="E27">A41</font></span>: 
 <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_6_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_6_1_1"/>
 for <font color="Olive" title="b1">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font> holds <br/><font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_6_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font> implies <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_6_1_1_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">O</font> being   <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">TT</font><span class="kw"> such that </span><br/><a NAME="E3:33_1_6_1_1_1"/><span class="lab"><font color="Green" title="E28">A42</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">O</font>
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_6_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:33_1_6_1_1_1_1"/>
( <font color="Maroon" title="c7">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c7">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_6_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_1_6_1_1_1_1_1"/>
<font color="Maroon" title="c7">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:33_1_6_1_1_1_1_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_6_1_1_1"><span class="lab"><font color="Green" title="E28">A42</font></span></a>, <a class="ref" href="pre_topc.html#T1" onmouseover="rs('pre_topc/T1')" onmouseout="rh()">PRE_TOPC:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_6_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_1_6_1_1_1_1_2"/>
 not <font color="Maroon" title="c7">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">O</font> = <font color="Maroon" title="c7">O</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E3:33_1_6_1_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A44</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">O</font> holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">b</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">O</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_6_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">O</font> implies  ex <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">b</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">O</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_6_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A45</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">O</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">b</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">O</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">x0</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font> ;<br/>
<a NAME="E3:33_1_6_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A46</font></span>: 
<font color="Maroon" title="c8">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c10">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_6_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E30">A45</font></span></a>, <a class="ref" href="fintopo7.html#D1" target="_self" onmouseover="rs('fintopo7/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">Z</font> =  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <span class="p1">(<span class="default"><a href="fintopo7.html#K3" title="FINTOPO7:func.3">gen_filter</a> <font color="Maroon" title="c3">TMP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x0</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c10">x0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E5:33_1_6_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c12">x1</font></span>)</span></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_6_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E31">A46</font></span></a>, <a class="ref" href="fintopo7.html#D9" target="_self" onmouseover="rs('fintopo7/D9')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c12">x1</font><span class="kw"> such that </span><br/><a NAME="E7:33_1_6_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E32">A48</font></span>: 
<font color="Maroon" title="c13">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">O</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cardfil2.html#D8" onmouseover="rs('cardfil2/D8')" onmouseout="rh()">CARDFIL2:def 8</a></span>;<br/>
<a NAME="E8:33_1_6_1_1_1_1_2_1"/>
(  not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c12">x1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c13">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c12">x1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<a NAME="E9:33_1_6_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c12">x1</font>
 
;<br/>
<a NAME="E10:33_1_6_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">b0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E12:33_1_6_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E33">A49</font></span>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b0</font>
 <span class="kw">and </span><br/><a NAME="E13:33_1_6_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E34">A50</font></span>: 
<font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">b0</font>
 ;<br/>
<a NAME="E14:33_1_6_1_1_1_1_2_1"/>
<font color="Maroon" title="c14">b0</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E15:33_1_6_1_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">b</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">O</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_1_6_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="txt" href="#E12:33_1_6_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A49</font></span></a>, <a class="txt" href="#E13:33_1_6_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E34">A50</font></span></a>, <a class="ref" href="connsp_2.html#T3" onmouseover="rs('connsp_2/T3')" onmouseout="rh()">CONNSP_2:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E4:33_1_6_1_1_1_1_2"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_6_1_1_1_1_2"><span class="lab"><font color="Green" title="E29">A44</font></span></a>, <a class="ref" href="connsp_2.html#T7" onmouseover="rs('connsp_2/T7')" onmouseout="rh()">CONNSP_2:7</a>, <a class="txt" href="#E3:33_1_6_1_1_1"><span class="lab"><font color="Green" title="E28">A42</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:33_1_6_1_1"/>
 <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:33_1_6_1"/>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_6_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_6_1_2"/><span class="lab"><font color="Green" title="E28">A51</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c6">t</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">TT</font> ;<br/>
<a NAME="E3:33_1_6_1_2"/>
<font color="Maroon" title="c7">t</font> is  <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_6_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_1_6_1_2_1"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">t</font> holds <br/><font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_6_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">ET</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">t</font> implies <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c8">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_6_1_2_1_1"/><span class="lab"><font color="Green" title="E29">A53</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">t</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c8">x</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">Z</font> =  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <span class="p1">(<span class="default"><a href="fintopo7.html#K3" title="FINTOPO7:func.3">gen_filter</a> <font color="Maroon" title="c3">TMP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x0</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E3:33_1_6_1_2_1_1"/><span class="lab"><font color="Green" title="E30">A55</font></span>: 
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cardfil2.html#K3" title="CARDFIL2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c10">x0</font></span>)</span></span><a href="cardfil2.html#K3" title="CARDFIL2:func.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D9" target="_self" onmouseover="rs('fintopo7/D9')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E4:33_1_6_1_2_1_1"/>
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">FX0</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">x0</font>) as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TMP</font> ;<br/>
<a NAME="E6:33_1_6_1_2_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_6_1_2"><span class="lab"><font color="Green" title="E28">A51</font></span></a>, <a class="txt" href="#E1:33_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E29">A53</font></span></a></span>;<br/>
<a NAME="E7:33_1_6_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c10">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:33_1_6_1_2_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E30">A55</font></span></a>, <a class="ref" href="cardfil2.html#D8" onmouseover="rs('cardfil2/D8')" onmouseout="rh()">CARDFIL2:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:33_1_6_1_2_1"/>
<font color="Maroon" title="c7">t</font> is  <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:33_1_6_1_2"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:33_1_6_1"/>
 <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_6_1"><span class="lab"><font color="Green" title="E27">A41</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:33_1_6"/>
 ex <font color="Olive" title="b1">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:33_1"/>
 ex <font color="Olive" title="b1">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">ET</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fintopo2.html#V1" title="FINTOPO2:attr.1">strict</a>   <a href="fintopo2.html#L1" title="FINTOPO2:struct.1">FMT_Space_Str</a> <span class="kw"> such that </span><br/><a NAME="E3:33"/><span class="lab"><font color="Green" title="E21">A56</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">ET</font>
 <span class="kw">and </span><br/><a NAME="E4:33"/>
( <font color="Maroon" title="c2">ET</font> is  <a href="fintopo7.html#V2" title="FINTOPO7:attr.2">U_FMT_filter</a>  &amp; <font color="Maroon" title="c2">ET</font> is  <a href="fintopo7.html#V3" title="FINTOPO7:attr.3">U_FMT_with_point</a>  &amp; <font color="Maroon" title="c2">ET</font> is  <a href="fintopo7.html#V4" title="FINTOPO7:attr.4">U_FMT_local</a>  )
 <span class="kw">and </span><br/><a NAME="E5:33"/><span class="lab"><font color="Green" title="E22">A57</font></span>: 
 ex <font color="Olive" title="b1">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>( <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">TT</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a><span class="kw"> such that </span><br/><a NAME="E7:33"/><span class="lab"><font color="Green" title="E23">A58</font></span>: 
<font color="Maroon" title="c3">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ET</font>
 <span class="kw">and </span><br/><a NAME="E8:33"/>
 <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c3">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E22">A57</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">TT</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TT</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c3">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<span class="kw">thus </span><a NAME="E9:33"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TT</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c3">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E21">A56</font></span></a>, <a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E22">A57</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E23">A58</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
