<?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="firebrick">::  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">Y1</font>, <font color="Olive" title="b4">Y2</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="b1">X1</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b2">X2</font> &amp; <font color="Olive" title="b2">X2</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">X1</font> &amp; <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b3">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Olive" title="b4">Y2</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><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></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">X1</font> &amp; <font color="Olive" title="b4">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><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></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b2">X2</font> holds <br/>( <font color="Olive" title="b3">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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> &amp; <font color="Olive" title="b4">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c4">Y1</font>, <font color="Maroon" title="c5">Y2</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="c2">X1</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c3">X2</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</font> &amp; <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X2</font> implies ( <font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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> &amp; <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c2">X1</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X2</font>
 <span class="kw">and </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c3">X2</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</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="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> or  not <font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</font> or  not <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X2</font> or ( <font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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> &amp; <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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><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="c2">X1</font>, <font color="Maroon" title="c7">A2</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c8">C1</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y1</font>, <font color="Maroon" title="c9">C2</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Y2</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" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:33"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</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="c4">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</font> or  not <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X2</font> or ( <font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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> &amp; <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:33"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</font>
 <span class="kw">and </span><br/><a NAME="E6:33"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <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> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X2</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="c4">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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> &amp; <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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><br/>

<a NAME="E7:33"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
 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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a></span>;<br/>
<a NAME="E8:33"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
 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">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E32">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_1"/>
<font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c9">C2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="tsep_1.html#D3" title="TSEP_1:def.3">TSEP_1:def 3</a></span>;<br/><a NAME="E3:33_1"/>
<font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A10</font></span>: <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">C2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">C2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span>

;<br/><a NAME="E5:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">C1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E6:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="tsep_1.html#D3" title="TSEP_1:def.3">TSEP_1:def 3</a></span>;<br/><a NAME="E7:33_1"/><span class="kw">then </span>
 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">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="tsep_1.html#D4" title="TSEP_1:def.4">TSEP_1:def 4</a></span>;<br/><a NAME="E8:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></span>;<br/><a NAME="E9:33_1"/>
<font color="Maroon" title="c7">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E10:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:33_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_2"/>
<font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c8">C1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="tsep_1.html#D3" title="TSEP_1:def.3">TSEP_1:def 3</a></span>;<br/><a NAME="E3:33_2"/>
<font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A13</font></span>: <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">C2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">C2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">C2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c9">C2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span>

;<br/><a NAME="E5:33_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">C2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E6:33_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="tsep_1.html#D3" title="TSEP_1:def.3">TSEP_1:def 3</a></span>;<br/><a NAME="E7:33_2"/><span class="kw">then </span>
 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="c5">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="tsep_1.html#D4" title="TSEP_1:def.4">TSEP_1:def 4</a></span>;<br/><a NAME="E8:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></span>;<br/><a NAME="E9:33_2"/>
<font color="Maroon" title="c6">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E10:33_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_2"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:33_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</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="E11:33"/>
( <font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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> &amp; <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E32">A8</font></span></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>
