<?xml version="1.0"?>
<div class="add">

<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">X1</font>, <font color="Olive" title="b2">X2</font>, <font color="Olive" title="b3">Y</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> holds <br/> ( ( <font color="Olive" title="b1">X1</font>,<font color="Olive" title="b3">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Olive" title="b2">X2</font>,<font color="Olive" title="b3">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies <font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Olive" title="b2">X2</font>,<font color="Olive" title="b3">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  ) &amp; ( <font color="Olive" title="b3">Y</font>,<font color="Olive" title="b1">X1</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Olive" title="b3">Y</font>,<font color="Olive" title="b2">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies <font color="Olive" title="b3">Y</font>,<font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Olive" title="b2">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</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> holds <br/> ( ( <font color="Maroon" title="c2">X1</font>,<font color="Olive" title="b1">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Maroon" title="c3">X2</font>,<font color="Olive" title="b1">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font>,<font color="Olive" title="b1">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  ) &amp; ( <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  ) )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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="c5">A1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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">let </span><font color="Maroon" title="c6">Y</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="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  ) &amp; ( <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</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="c6">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">thus </span><a NAME="E4:110"/>
( <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">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="c6">Y</font>,<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1"/>
( <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">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">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a> </span><br/>

<a NAME="E2:110_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A1</font></span>: 
( <font color="Maroon" title="c5">A1</font>,<font color="Maroon" title="c7">C</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  &amp; <font color="Maroon" title="c4">A2</font>,<font color="Maroon" title="c7">C</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#D9" target="_self" title="TSEP_1:def.9">Def9</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">D</font>, <font color="Maroon" title="c9">C</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="c8">D</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="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font></span>)</span> &amp; <font color="Maroon" title="c9">C</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 <font color="Maroon" title="c6">Y</font> implies <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">C</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:110_1_1"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c8">D</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="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:110_1_1"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon" title="c9">C</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 <font color="Maroon" title="c6">Y</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="c8">D</font>,<font color="Maroon" title="c9">C</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> </span><br/><a NAME="E3:110_1_1"/>
<font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="tsep_1.html#D2" target="_self" title="TSEP_1:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:110_1_1"/>
<font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">C</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="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="tsep_1.html#T58" target="_self" title="TSEP_1:th.58">Th58</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">hence </span><a NAME="E4:110_1"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c6">Y</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D9" target="_self" title="TSEP_1:def.9">Def9</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">hence </span><a NAME="E5:110"/>
( <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  &amp; <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">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"> verum</span><br/>


</div>
