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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font>, <font color="Olive" title="b3">C1</font>, <font color="Olive" title="b4">C2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b3">C1</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp; <font color="Olive" title="b2">A2</font>,<font color="Olive" title="b4">C2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  holds <br/>( <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  iff <font color="Olive" title="b3">C1</font>,<font color="Olive" title="b4">C2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font>, <font color="Maroon" title="c4">C1</font>, <font color="Maroon" title="c5">C2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">C1</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp; <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c5">C2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  implies ( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  iff <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c5">C2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:16"/>
( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">C1</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp; <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c5">C2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  iff <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c5">C2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  )</span><br/>

<a NAME="E2:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A1</font></span>: 
( <font color="Maroon" title="c4">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_2.html#T4" target="_self" title="TSEP_2:th.4">Th4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:16"/>
( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  implies <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c5">C2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c5">C2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  implies <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1"/>
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c3">A2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tsep_1.html#D7" title="TSEP_1:def.7">TSEP_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c5">C2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> </span><br/>

<a NAME="E2:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">C1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="tsep_2.html#T1" target="_self" title="TSEP_2:th.1">Th1</a></span>;<br/>
<a NAME="E3:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">C2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_2.html#T1" target="_self" title="TSEP_2:th.1">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:16_1"/>
<font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c5">C2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D7" title="TSEP_1:def.7">TSEP_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E4:16"/>
<font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c5">C2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> </span><br/>

<a NAME="E5:16"/><span class="kw">then </span>
<font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">C2</font>,<font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">C1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D7" title="TSEP_1:def.7">TSEP_1:def 7</a></span>;<br/>
<a NAME="E6:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">C1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_2.html#T1" target="_self" title="TSEP_2:th.1">Th1</a></span>;<br/>
<a NAME="E7:16"/><span class="kw">then </span>
<font color="Maroon" title="c3">A2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="tsep_2.html#T1" target="_self" title="TSEP_2:th.1">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:16"/>
<font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D7" title="TSEP_1:def.7">TSEP_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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