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<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">X0</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> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Olive" title="b2">X2</font> holds <br/>( ( <font color="Olive" title="b1">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b3">X0</font> implies ( <font color="Olive" title="b3">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b2">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Olive" title="b1">X1</font> &amp; <font color="Olive" title="b2">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b3">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Olive" title="b1">X1</font> ) ) &amp; ( <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">X0</font> implies ( <font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b3">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Olive" title="b2">X2</font> &amp; <font color="Olive" title="b3">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b1">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</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">X0</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> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> implies ( ( <font color="Maroon" title="c2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X0</font> implies ( <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font> ) ) &amp; ( <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">X0</font> implies ( <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> ) ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:31"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</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"> ( ( <font color="Maroon" title="c2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X0</font> implies ( <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font> ) ) &amp; ( <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">X0</font> implies ( <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> ) ) )</span><br/>

<div><span class="lab"><font color="Green" title="E24">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:31_1"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</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="c3">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tsep_1.html#T27" title="TSEP_1:th.27">TSEP_1:27</a></span>;<br/><a NAME="E2:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T17" target="_self" title="TMAP_1:th.17">Th22</a></span>;<br/><span class="kw">assume </span><a NAME="E3:31_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<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">X0</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> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> )</span><br/><a NAME="E4:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</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="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tmap_1.html#T27" target="_self" title="TMAP_1:th.27">Th32</a></span>;<br/><span class="kw">hence </span><a NAME="E5:31_1"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="tmap_1.html#T18" target="_self" title="TMAP_1:th.18">Th23</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="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font></span><br/><a NAME="E6:31_1"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c4">X0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="tmap_1.html#T18" target="_self" title="TMAP_1:th.18">Th23</a></span>;<br/><span class="kw">hence </span><a NAME="E7:31_1"/>
<font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="tsep_1.html#T26" title="TSEP_1:th.26">TSEP_1:26</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="31_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:31_2"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</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="c2">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tsep_1.html#T27" title="TSEP_1:th.27">TSEP_1:27</a></span>;<br/><a NAME="E2:31_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T17" target="_self" title="TMAP_1:th.17">Th22</a></span>;<br/><span class="kw">assume </span><a NAME="E3:31_2"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X0</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">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font> )</span><br/><a NAME="E4:31_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</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">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tmap_1.html#T27" target="_self" title="TMAP_1:th.27">Th32</a></span>;<br/><span class="kw">hence </span><a NAME="E5:31_2"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_2"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="tmap_1.html#T18" target="_self" title="TMAP_1:th.18">Th23</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="c3">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font></span><br/><a NAME="E6:31_2"/>
<font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="tmap_1.html#T18" target="_self" title="TMAP_1:th.18">Th23</a></span>;<br/><span class="kw">hence </span><a NAME="E7:31_2"/>
<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_2"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="tsep_1.html#T26" title="TSEP_1:th.26">TSEP_1:26</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="E4:31"/>
( ( <font color="Maroon" title="c2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X0</font> implies ( <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font> ) ) &amp; ( <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">X0</font> implies ( <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c4">X0</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E24">A2</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/>


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