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<b>let </b><font color="Maroon" title="c1">FMT</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fintopo2.html#L1" title="FINTOPO2:struct.1">FMT_Space_Str</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font><br/> for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b2">V1</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</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="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b4">W</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) ) iff for <font color="Olive" title="b1">A</font>, <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">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )</span><br/>

<b>thus </b><a NAME="E1:41"/>
( ( 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">FMT</font><br/> for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b2">V1</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</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="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b4">W</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) ) implies for <font color="Olive" title="b1">A</font>, <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">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">A</font>, <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">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  ) implies 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">FMT</font><br/> for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b2">V1</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</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="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b4">W</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E17">A1</font></i>: 
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">FMT</font><br/> for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b2">V1</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</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="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b4">W</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <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">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span><br/>

<b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span><br/>

<b>thus </b><a NAME="E2:41_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_1_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="c1">FMT</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )</span><br/>

<a NAME="E1:41_1_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span><br/>

<a NAME="E2:41_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:41_1_1_1_1"/><b>then </b>
(  ex <font color="Olive" title="b1">W1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b1">W1</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">x</font> &amp; <font color="Olive" title="b1">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> ) &amp;  ex <font color="Olive" title="b1">W2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b1">W2</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">x</font> &amp; <font color="Olive" title="b1">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> ) )
 
<b>by </b><i><a class="ref" href="fintopo2.html#T22" target="_self" title="FINTOPO2:th.22">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">W1</font>, <font color="Maroon" title="c6">W2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font><b> such that </b><br/><a NAME="E5:41_1_1_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c5">W1</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">x</font> &amp; <font color="Maroon" title="c6">W2</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">x</font> )
 <b>and </b><br/><a NAME="E6:41_1_1_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c5">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font><b> such that </b><br/><a NAME="E8:41_1_1_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c7">W</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">x</font>
 <b>and </b><br/><a NAME="E9:41_1_1_1_1"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">W2</font>
 <b>by </b><i><a class="txt" href="#E1:41_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E5:41_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>
<a NAME="E10:41_1_1_1_1"/>
( <font color="Maroon" title="c5">W1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W1</font> &amp; <font color="Maroon" title="c5">W1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:41_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W1</font> &amp; <font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W2</font> )
 
<b>by </b><i><a class="txt" href="#E9:41_1_1_1_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:41_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">W</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">x</font> &amp; <font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="#E6:41_1_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E8:41_1_1_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:41_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">W</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">x</font> &amp; <font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<b>hence </b><a NAME="E14:41_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:41_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1_1_2"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span></span><br/>

<a NAME="E2:41_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="fintopo2.html#T30" target="_self" title="FINTOPO2:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E1:41_1_1_1_2"><i><font color="Green" title="E19">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:41_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )
 <b>by </b><i><a class="txt" href="#E1:41_1_1_1"><i><font color="Green" title="E18">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:41_1"/>
verum
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:41"/>
( ( for <font color="Olive" title="b1">A</font>, <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">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  ) implies 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">FMT</font><br/> for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b2">V1</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</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="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b4">W</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<a NAME="E1:41_2"/>
(  ex <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">FMT</font> ex <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b2">V1</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</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="Olive" title="b1">x</font> &amp; ( for <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b4">W</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="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) ) implies  ex <font color="Olive" title="b1">A</font>, <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">FMT</font> st <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c2">x0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font>, <font color="Maroon" title="c3">V1</font>, <font color="Maroon" title="c4">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font><b> such that </b><a NAME="E2:41_2_1"/><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c3">V1</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="c2">x0</font> &amp; <font color="Maroon" title="c4">V2</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="c2">x0</font> )
 <b>and </b><br/><a NAME="E3:41_2_1"/><i><font color="Green" title="E18">A9</font></i>: 
for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b1">W</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="c2">x0</font> holds <br/> not <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">A</font>, <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">FMT</font> st <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span><br/>

<a NAME="E4:41_2_1"/>
( <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  &amp; <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V2</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )
 
<b>by </b><i><a class="txt" href="#E2:41_2_1"><i><font color="Green" title="E17">A8</font></i></a></i>;<br/>
<a NAME="E5:41_2_1"/><b>then </b><i><font color="Green" title="E19">A10</font></i>: 
( <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> &amp; not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  )
 
<b>by </b><i><a class="txt" href="#E3:41_2_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="fintopo2.html#T22" target="_self" title="FINTOPO2:th.22">Th22</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">V1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">FMT</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span><br/>

<b>take </b>
<font color="Maroon" title="c4">V2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span><br/>

<b>thus </b><a NAME="E6:41_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> 
 <b>by </b><i><a class="txt" href="#E5:41_2_1"><i><font color="Green" title="E19">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_2"/>
( ( for <font color="Olive" title="b1">A</font>, <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">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  ) implies 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">FMT</font><br/> for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b2">V1</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</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="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b4">W</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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div>
