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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">K</font> is  <a href="simplex0.html#V2" title="SIMPLEX0:attr.2">finite-vertices</a> ;<br/>
<a NAME="E1:37_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D10" title="SIMPLEX0:def.10">SIMPLEX0:def 10</a></span>;<br/>
<a NAME="E2:37_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">K</font>
 
;<br/>
<a NAME="E3:37_1_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1_1_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<a NAME="E2:37_1_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D10" title="SIMPLEX0:def.10">SIMPLEX0:def 10</a></span>;<br/>
<a NAME="E3:37_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">K</font></span>)</span></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">K</font></span>)</span>
 
;<br/>
 <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c3">K</font> = 
 <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="simplex1.html#T20" target="_self" title="SIMPLEX1:th.20">Th20</a></span>
<br/>.= 
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <span class="p1">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="simplex1.html#D5" target="_self" title="SIMPLEX1:def.5">Def5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:37_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_1"><span class="lab"><font color="Green" title="E29">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:37_1_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="simplex1.html#T16" target="_self" title="SIMPLEX1:th.16">Th16</a></span>;<br/>
<a NAME="E5:37_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:37_1_1"/>
 <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">K</font> is  <a href="simplex0.html#V2" title="SIMPLEX0:attr.2">finite-vertices</a> 
 ; <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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