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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font>, <font color="Olive" title="b2">X1</font>, <font color="Olive" title="b3">X2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b3">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  holds <br/>( <font color="Olive" title="b2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b1">Y</font>,<font color="Olive" title="b3">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b1">Y</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  &amp; <font color="Olive" title="b1">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b2">X1</font>,<font color="Olive" title="b1">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b3">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  implies ( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  &amp; <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:86"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:86"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  or ( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  &amp; <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">A2</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">A1</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">C</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:86"/>
<font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  &amp; <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  )</span><br/>

<a NAME="E7:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c6">A1</font>,<font color="Maroon" title="c5">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="ref" href="tsep_1.html#D6" target="_self" title="TSEP_1:def.6">Def8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">D1</font>, <font color="Maroon" title="c9">D2</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">D1</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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp; <font color="Maroon" title="c9">D2</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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font></span>)</span> implies <font color="Maroon" title="c8">D1</font>,<font color="Maroon" title="c9">D2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:86_1"/>
( <font color="Maroon" title="c8">D1</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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp; <font color="Maroon" title="c9">D2</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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">D1</font>,<font color="Maroon" title="c9">D2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><br/><a NAME="E2:86_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D1</font> &amp; <font color="Maroon" title="c5">A2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">D2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="tsep_1.html#D4" target="_self" title="TSEP_1:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:86_1"/>
<font color="Maroon" title="c8">D1</font>,<font color="Maroon" title="c9">D2</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="#E7:86"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="tsep_1.html#T39" target="_self" title="TSEP_1:th.39">Th43</a></span>; <a class="txt" 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="E9:86"/>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D6" target="_self" title="TSEP_1:def.6">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> </span><br/>

<a NAME="E10:86"/><span class="kw">then </span>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="tsep_1.html#T26" target="_self" title="TSEP_1:th.26">Th29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:86"/>
<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="tsep_1.html#T26" target="_self" title="TSEP_1:th.26">Th29</a></span>; <a class="txt" 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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