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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="jordan2c.html#V1" title="JORDAN2C:attr.1">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="E1:133"/><span class="lab"><font color="Green" title="E101">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="jordan2c.html#V1" title="JORDAN2C:attr.1">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">P</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">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="E2:133"/><span class="lab"><font color="Green" title="E102">A2</font></span>: 
<font color="Maroon" title="c4">P</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 <span class="kw">and </span><br/><a NAME="E3:133"/><span class="lab"><font color="Green" title="E103">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="E4:133"/><span class="lab"><font color="Green" title="E104">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E5:133"/><span class="lab"><font color="Green" title="E105">A5</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">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="#E4:133"><span class="lab"><font color="Green" title="E104">A4</font></span></a>, <a class="ref" href="jordan2c.html#D4" target="_self" title="JORDAN2C:def.4">Def4</a></span>;<br/>
<a NAME="E6:133"/>
 <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="#E1:133"><span class="lab"><font color="Green" title="E101">A1</font></span></a>, <a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E102">A2</font></span></a>, <a class="ref" href="jordan2c.html#T76" target="_self" title="JORDAN2C:th.76">Th76</a></span>;<br/>
<a NAME="E7:133"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">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#R4" title="CONNSP_1:pred.4">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="ref" href="jordan2c.html#D4" target="_self" title="JORDAN2C:def.4">Def4</a></span>;<br/>
<a NAME="E8:133"/><span class="lab"><font color="Green" title="E107">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="#E1:133"><span class="lab"><font color="Green" title="E101">A1</font></span></a>, <a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E102">A2</font></span></a>, <a class="ref" href="jordan2c.html#T74" target="_self" title="JORDAN2C:th.74">Th74</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:133"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">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="#E5:133"><span class="lab"><font color="Green" title="E105">A5</font></span></a>, <a class="ref" href="sprect_1.html#T6" title="SPRECT_1:th.6">SPRECT_1:6</a></span>;<br/>
<a NAME="E10:133"/><span class="lab"><font color="Green" title="E109">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="#E4:133"><span class="lab"><font color="Green" title="E104">A4</font></span></a>, <a class="ref" href="jordan2c.html#T27" target="_self" title="JORDAN2C:th.27">Th27</a></span>;<br/>
<a NAME="E11:133"/><span class="lab"><font color="Green" title="E110">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="#E3:133"><span class="lab"><font color="Green" title="E103">A3</font></span></a>, <a class="ref" href="jordan2c.html#T27" target="_self" title="JORDAN2C:th.27">Th27</a></span>;<br/>
<a NAME="E12:133"/><span class="lab"><font color="Green" title="E111">A11</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">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="#E3:133"><span class="lab"><font color="Green" title="E103">A3</font></span></a>, <a class="ref" href="jordan2c.html#D4" target="_self" title="JORDAN2C:def.4">Def4</a></span>;<br/>
<a NAME="E13:133"/><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="#E8:133"><span class="lab"><font color="Green" title="E107">A7</font></span></a>, <a class="ref" href="sprect_1.html#T6" title="SPRECT_1:th.6">SPRECT_1:6</a></span>;<br/>
<a NAME="E14:133"/><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="#E12:133"><span class="lab"><font color="Green" title="E111">A11</font></span></a>, <a class="txt" href="#E7:133"><span class="lab"><font color="Green" title="E106">A6</font></span></a>, <a class="txt" href="#E11:133"><span class="lab"><font color="Green" title="E110">A10</font></span></a>, <a class="ref" href="goboard9.html#T3" title="GOBOARD9:th.3">GOBOARD9:3</a>, <a class="ref" href="xboole_1.html#T69" title="XBOOLE_1:th.69">XBOOLE_1:69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:133"/>
<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="#E5:133"><span class="lab"><font color="Green" title="E105">A5</font></span></a>, <a class="txt" href="#E9:133"><span class="lab"><font color="Green" title="E108">A8</font></span></a>, <a class="txt" href="#E7:133"><span class="lab"><font color="Green" title="E106">A6</font></span></a>, <a class="txt" href="#E10:133"><span class="lab"><font color="Green" title="E109">A9</font></span></a>, <a class="ref" href="goboard9.html#T3" title="GOBOARD9:th.3">GOBOARD9:3</a>, <a class="ref" href="xboole_1.html#T69" title="XBOOLE_1:th.69">XBOOLE_1:69</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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