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<span class="kw">let </span><font color="Maroon" title="c1">ET</font> be   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="fintopo7.html#V2" title="FINTOPO7:attr.2">U_FMT_filter</a>   <a href="fintopo2.html#L1" title="FINTOPO2:struct.1">FMT_Space_Str</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">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font><br/>  for <font color="Olive" title="b2">NA</font> being    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Olive" title="b1">A</font><br/>  for <font color="Olive" title="b3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>  st <font color="Olive" title="b2">NA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> holds <br/><font color="Olive" title="b3">B</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">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">NA</font> being    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font><br/>  for <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">ET</font>  st <font color="Olive" title="b1">NA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b2">B</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">NA</font> be    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</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">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>  st <font color="Maroon" title="c3">NA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b1">B</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">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="c3">NA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c4">B</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c3">NA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</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">B</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font></span><br/>

<a NAME="E2:17"/>
 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="c1">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="c2">A</font> holds <br/><font color="Maroon" title="c4">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="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_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="c1">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="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c4">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="c5">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1"/>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</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">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="c5">x</font></span><br/>

<a NAME="E2:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c3">NA</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="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D6" target="_self" onmouseover="rs('fintopo7/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E3:17_1"/>
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c5">x</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D2" target="_self" onmouseover="rs('fintopo7/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:17_1"/>
<font color="Maroon" title="c4">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="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="card_fil.html#D1" onmouseover="rs('card_fil/D1')" onmouseout="rh()">CARD_FIL:def 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>
<span class="kw">hence </span><a NAME="E3:17"/>
<font color="Maroon" title="c4">B</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D6" target="_self" onmouseover="rs('fintopo7/D6')" onmouseout="rh()">Def6</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/>


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