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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">X1</font>, <font color="Olive" title="b2">Y1</font>, <font color="Olive" title="b3">X2</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> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Olive" title="b2">Y1</font> &amp; <font color="Olive" title="b1">X1</font>,<font color="Olive" title="b3">X2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  &amp; <font color="Olive" title="b2">Y1</font>,<font color="Olive" title="b4">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  holds <br/><font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b2">Y1</font>,<font color="Olive" title="b3">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Olive" title="b4">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">Y1</font>, <font color="Maroon" title="c4">X2</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="Red">::  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">Y1</font> &amp; <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  &amp; <font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c5">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  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="c3">Y1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</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="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="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" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">B1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y1</font>, <font color="Maroon" title="c9">B2</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="E3:39"/><span class="lab"><font color="Green" title="E27">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">Y1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  or  not <font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c5">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  or <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">Y1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:39"/>
for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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> <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="c2">X1</font> &amp; <font color="Olive" title="b2">A2</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="c4">X2</font> holds <br/><font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tsep_2.html#D2" target="_self" title="TSEP_2:def.2">TSEP_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c5">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  or <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">Y1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  )</span><br/>

<a NAME="E5:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c6">A1</font>,<font color="Maroon" title="c7">A2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 
;<br/>
<span class="kw">assume </span><a NAME="E6:39"/>
for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</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">B1</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="c3">Y1</font> &amp; <font color="Olive" title="b2">B2</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="c5">Y2</font> holds <br/><font color="Olive" title="b1">B1</font>,<font color="Olive" title="b2">B2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tsep_2.html#D2" target="_self" title="TSEP_2:def.2">TSEP_2:def 2</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#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a> </span><br/>

<a NAME="E7:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c8">B1</font>,<font color="Maroon" title="c9">B2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">C</font>, <font color="Maroon" title="c11">D</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="c10">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 <span class="p1">(<span class="default"><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">Y1</font></span>)</span> &amp; <font color="Maroon" title="c11">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="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font></span>)</span> implies <font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c11">D</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:39_1"/>
( <font color="Maroon" title="c10">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 <span class="p1">(<span class="default"><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">Y1</font></span>)</span> &amp; <font color="Maroon" title="c11">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="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font></span>)</span> )
 ; <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="c10">C</font>,<font color="Maroon" title="c11">D</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> </span><br/><a NAME="E2:39_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B1</font> &amp; <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">B2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a>, <a class="ref" href="tsep_1.html#D5" title="TSEP_1:def.5">TSEP_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_1"/>
<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c11">D</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="tsep_2.html#T14" target="_self" title="TSEP_2:th.14">Th14</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="E9:39"/>
<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">Y1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_2.html#D2" target="_self" title="TSEP_2:def.2">Def2</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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