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<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is <a href="frechet.html#V3" title="FRECHET:attr.3">Frechet</a> 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">T</font> holds <br/> (  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font> &amp;  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b2">B</font></span>)</span> &amp;  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font> ) )</span><br/>

<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="frechet.html#V3" title="FRECHET:attr.3">Frechet</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">T</font> holds <br/> (  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font> &amp;  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b2">B</font></span>)</span> &amp;  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font> )</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">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c3">B</font></span>)</span> &amp;  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>thus </b><a NAME="E2:37"/>
(  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c3">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="frechet2.html#T20" target="_self" title="FRECHET2:th.20">Th20</a>, <a class="ref" href="frechet2.html#T21" target="_self" title="FRECHET2:th.21">Th21</a>, <a class="ref" href="frechet2.html#T22" target="_self" title="FRECHET2:th.22">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font></span><br/>

<b>thus </b> <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font></span>)</span> = 
 <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="frechet2.html#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="frechet2.html#T23" target="_self" title="FRECHET2:th.23">Th23</a></i>
<br/>.= 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="frechet2.html#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="frechet2.html#T23" target="_self" title="FRECHET2:th.23">Th23</a></i>
<br/>.= 
 <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="frechet2.html#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="frechet2.html#T23" target="_self" title="FRECHET2:th.23">Th23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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