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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies for <font color="Olive" title="b1">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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">P</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> holds <br/> not <font color="Olive" title="b1">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> )</span><br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">P</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> holds <br/> not <font color="Olive" title="b1">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a></span><br/>

<b>let </b><font color="Maroon" title="c2">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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> implies  not <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> )</span><br/>

<a NAME="E2:8"/>
 <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">W</font> =  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<b>assume </b><a NAME="E4:8"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c2">P</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> &amp; <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>
<br/>.= 
<font color="Maroon" title="c3">W</font>

;<br/>
<b>hence </b><a NAME="E6:8"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E4:8"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="jordan2c.html#T39" title="JORDAN2C:th.39">JORDAN2C:39</a>, <a class="ref" href="tbsp_1.html#T20" title="TBSP_1:th.20">TBSP_1:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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