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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">P</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> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b3">P</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>  &amp; <font color="Olive" title="b1">A</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">B</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Olive" title="b3">P</font> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:126"/><span class="lab"><font color="Green" title="E89">A1</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">P</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> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b3">P</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>  &amp; <font color="Olive" title="b1">A</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">B</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Olive" title="b3">P</font> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">P</font> be   <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> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">P</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">B</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c4">P</font> implies <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:126"/><span class="lab"><font color="Green" title="E90">A2</font></span>: 
<font color="Maroon" title="c4">P</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E3:126"/><span class="lab"><font color="Green" title="E91">A3</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c4">P</font>
 <span class="kw">and </span><br/><a NAME="E4:126"/><span class="lab"><font color="Green" title="E92">A4</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c4">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font></span><br/>

<a NAME="E5:126"/><span class="lab"><font color="Green" title="E93">A5</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c4">P</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="#E4:126"><span class="lab"><font color="Green" title="E92">A4</font></span></a></span>;<br/>
<a NAME="E6:126"/>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c4">P</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:126"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="txt" href="#E2:126"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="ref" href="jordan2c.html#T68" target="_self" onmouseover="rs('jordan2c/T68')" onmouseout="rh()">Th53</a></span>;<br/>
<a NAME="E7:126"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A6</font></span>: 
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c4">P</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c4">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
;<br/>
<a NAME="E8:126"/><span class="lab"><font color="Green" title="E95">A7</font></span>: 
 not <font color="Maroon" title="c4">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:126"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="txt" href="#E2:126"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="ref" href="jordan2c.html#T66" target="_self" onmouseover="rs('jordan2c/T66')" onmouseout="rh()">Th51</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:126"/><span class="lab"><font color="Green" title="E96">A8</font></span>: 
<font color="Maroon" title="c3">B</font> <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="#E4:126"><span class="lab"><font color="Green" title="E92">A4</font></span></a></span>;<br/>
<a NAME="E10:126"/><span class="lab"><font color="Green" title="E97">A9</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:126"><span class="lab"><font color="Green" title="E92">A4</font></span></a>, <a class="ref" href="jordan2c.html#T23" target="_self" onmouseover="rs('jordan2c/T23')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E11:126"/><span class="lab"><font color="Green" title="E98">A10</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:126"><span class="lab"><font color="Green" title="E91">A3</font></span></a>, <a class="ref" href="jordan2c.html#T23" target="_self" onmouseover="rs('jordan2c/T23')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E12:126"/><span class="lab"><font color="Green" title="E99">A11</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c4">P</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="#E3:126"><span class="lab"><font color="Green" title="E91">A3</font></span></a></span>;<br/>
<a NAME="E13:126"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <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="#E8:126"><span class="lab"><font color="Green" title="E95">A7</font></span></a>, <a class="ref" href="sprect_1.html#T4" onmouseover="rs('sprect_1/T4')" onmouseout="rh()">SPRECT_1:4</a></span>;<br/>
<a NAME="E14:126"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:126"><span class="lab"><font color="Green" title="E99">A11</font></span></a>, <a class="txt" href="#E7:126"><span class="lab"><font color="Green" title="E94">A6</font></span></a>, <a class="txt" href="#E11:126"><span class="lab"><font color="Green" title="E98">A10</font></span></a>, <a class="ref" href="goboard9.html#T1" onmouseover="rs('goboard9/T1')" onmouseout="rh()">GOBOARD9:1</a>, <a class="ref" href="xboole_1.html#T69" onmouseover="rs('xboole_1/T69')" onmouseout="rh()">XBOOLE_1:69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:126"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:126"><span class="lab"><font color="Green" title="E93">A5</font></span></a>, <a class="txt" href="#E9:126"><span class="lab"><font color="Green" title="E96">A8</font></span></a>, <a class="txt" href="#E7:126"><span class="lab"><font color="Green" title="E94">A6</font></span></a>, <a class="txt" href="#E10:126"><span class="lab"><font color="Green" title="E97">A9</font></span></a>, <a class="ref" href="goboard9.html#T1" onmouseover="rs('goboard9/T1')" onmouseout="rh()">GOBOARD9:1</a>, <a class="ref" href="xboole_1.html#T69" onmouseover="rs('xboole_1/T69')" onmouseout="rh()">XBOOLE_1:69</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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