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<div><span class="kw">hereby </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="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">A2</font> &amp; <font color="Olive" title="b1">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">A2</font> &amp; <font color="Olive" title="b1">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b2">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) implies <font color="Maroon" title="c1">S</font> is  <a href="jordan1.html#V1" title="JORDAN1:attr.1">Jordan</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:113_1_1_1"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
<font color="Maroon" title="c1">S</font> is  <a href="jordan1.html#V1" title="JORDAN1:attr.1">Jordan</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="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">A2</font> &amp; <font color="Olive" title="b1">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">A2</font> &amp; <font color="Olive" title="b1">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b2">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) )</span><br/><span class="kw">hence </span><a NAME="E2:113_1_1_1"/>
<font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1.html#D2" title="JORDAN1:def.2">JORDAN1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">A2</font> &amp; <font color="Olive" title="b1">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">A2</font> &amp; <font color="Olive" title="b1">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b2">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><span class="kw">consider </span><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:113_1_1_1"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>
 <span class="kw">and </span><br/><a NAME="E5:113_1_1_1"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
<font color="Maroon" title="c2">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A2</font>
 <span class="kw">and </span><br/><a NAME="E6:113_1_1_1"/><span class="lab"><font color="Green" title="E85">A4</font></span>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A2</font>
 <span class="kw">and </span><br/><a NAME="E7:113_1_1_1"/><span class="lab"><font color="Green" title="E86">A5</font></span>: 
for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  st <font color="Olive" title="b1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Olive" title="b2">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font> holds <br/>( <font color="Olive" title="b1">C1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b2">C2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:113_1_1_1"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="jordan1.html#D2" title="JORDAN1:def.2">JORDAN1:def 2</a></span>;<br/><a NAME="E8:113_1_1_1"/><span class="lab"><font color="Green" title="E87">A6</font></span>: 
<font color="Maroon" title="c3">A2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:113_1_1_1"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></span>;<br/><a NAME="E9:113_1_1_1"/>
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:113_1_1_1"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">C1</font> = <font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c5">C2</font> = <font color="Maroon" title="c3">A2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:113_1_1_1"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="ref" href="tops_2.html#T38" title="TOPS_2:th.38">TOPS_2:38</a></span>;<br/><a NAME="E11:113_1_1_1"/>
<font color="Maroon" title="c5">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font>
 ;<br/><a NAME="E12:113_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A7</font></span>: 
<font color="Maroon" title="c4">C1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:113_1_1_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">A1</font> = <font color="Maroon" title="c2">A1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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="Olive" title="b1">A2</font> &amp; <font color="Maroon" title="c6">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">A2</font> &amp; <font color="Maroon" title="c6">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b1">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><span class="kw">take </span><font color="Maroon" title="c7">A2</font> = <font color="Maroon" title="c3">A2</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="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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> &amp; <font color="Maroon" title="c6">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A2</font> &amp; <font color="Maroon" title="c6">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><span class="kw">thus </span><a NAME="E13:113_1_1_1"/>
<font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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="txt" href="#E4:113_1_1_1"><span class="lab"><font color="Green" title="E83">A2</font></span></a></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="c6">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A2</font> &amp; <font color="Maroon" title="c6">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><span class="kw">thus </span><a NAME="E14:113_1_1_1"/>
<font color="Maroon" title="c6">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:113_1_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A2</font> &amp; <font color="Maroon" title="c6">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><span class="kw">thus </span><a NAME="E15:113_1_1_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a></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="c6">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><a NAME="E16:113_1_1_1"/>
<font color="Maroon" title="c4">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A1</font>
 ;<br/><a NAME="E17:113_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">C2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:113_1_1_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:113_1_1_1"/>
( <font color="Maroon" title="c6">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c7">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:113_1_1_1"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</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">assume </span><a NAME="E2:113_1_1"/><span class="lab"><font color="Green" title="E82">A8</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">A1</font>, <font color="Olive" title="b2">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">A2</font> or  not <font color="Olive" title="b1">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">A2</font> or  not <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">A2</font> or  not <font color="Olive" title="b1">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Olive" title="b2">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) or <font color="Maroon" title="c1">S</font> is  <a href="jordan1.html#V1" title="JORDAN1:attr.1">Jordan</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><a NAME="E4:113_1_1"/><span class="lab"><font color="Green" title="E83">A9</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>
 <span class="kw">and </span><br/><a NAME="E5:113_1_1"/><span class="lab"><font color="Green" title="E84">A10</font></span>: 
<font color="Maroon" title="c2">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A2</font>
 <span class="kw">and </span><br/><a NAME="E6:113_1_1"/><span class="lab"><font color="Green" title="E85">A11</font></span>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A2</font>
 <span class="kw">and </span><br/><a NAME="E7:113_1_1"/><span class="lab"><font color="Green" title="E86">A12</font></span>: 
<font color="Maroon" title="c2">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">and </span><br/><a NAME="E8:113_1_1"/><span class="lab"><font color="Green" title="E87">A13</font></span>: 
<font color="Maroon" title="c3">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</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="c1">S</font> is  <a href="jordan1.html#V1" title="JORDAN1:attr.1">Jordan</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">a1</font> = <font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c5">a2</font> = <font color="Maroon" title="c3">A2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E10:113_1_1"/><span class="lab"><font color="Green" title="E88">A14</font></span>: 
 ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Olive" title="b1">B</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:113_1_1"><span class="lab"><font color="Green" title="E87">A13</font></span></a>, <a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:113_1_1"/>
<font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E2:113_1_1"><span class="lab"><font color="Green" title="E82">A8</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="jordan1.html#D2" title="JORDAN1:def.2">JORDAN1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or ( <font color="Olive">b<sub>3</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> ) ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c4">a1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c4">a1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or ( <font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> ) ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">a2</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="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">a2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">a2</font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font> or ( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E12:113_1_1"/>
<font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:113_1_1"><span class="lab"><font color="Green" title="E83">A9</font></span></a></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="c4">a1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">a2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">a2</font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font> or ( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:113_1_1"/>
<font color="Maroon" title="c4">a1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E5:113_1_1"><span class="lab"><font color="Green" title="E84">A10</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">a2</font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font> or ( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E14:113_1_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1_1"><span class="lab"><font color="Green" title="E85">A11</font></span></a></span>; <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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font> or ( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> ) )</span><br/>

<a NAME="E15:113_1_1"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Olive" title="b1">B</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:113_1_1"><span class="lab"><font color="Green" title="E86">A12</font></span></a>, <a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:113_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font> or ( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:113_1_1"><span class="lab"><font color="Green" title="E88">A14</font></span></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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