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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:122_1_1_1"/>
( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:122_1_1_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</span>)</span>),<font color="Maroon" title="c1">K</font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font></span><br/><div class="add"><a NAME="E2:122_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">F</font> =  the    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1,<font color="Maroon" title="c1">K</font> as    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</span>)</span>),<font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font></span><br/><span class="kw">thus </span><a NAME="E4:122_1_1_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:122_1_1_1_2"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</span>)</span>),<font color="Maroon" title="c1">K</font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font></span><br/><div class="add"><a NAME="E2:122_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A4</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:122"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="simplex0.html#D18" target="_self" title="SIMPLEX0:def.18">Def18</a></span>;<br/><a NAME="E3:122_1_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">S</font> 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="#E1:122_1_1_1_2"><span class="lab"><font color="Green" title="E48">A3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:122_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">I</font> = <font color="Maroon" title="c2">i</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E1:122_1_1_1_2"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><a NAME="E7:122_1_1_1_2"/>
<font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <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="#E2:122"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="xxreal_3.html#T37" title="XXREAL_3:th.37">XXREAL_3:37</a></span>;<br/><a NAME="E8:122_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">Sx</font> = <font color="Maroon" title="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c1">K</font> ;<br/><a NAME="E10:122_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">Sx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">I</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1</span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:122_1_1_1_2"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E5:122_1_1_1_2"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="stirl2_1.html#T55" title="STIRL2_1:th.55">STIRL2_1:55</a></span>;<br/><a NAME="E11:122_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122_1_1_1_2"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E12:122_1_1_1_2"/><span class="kw">then </span>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">Sx</font> = <font color="Maroon" title="c6">Sx</font> as    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</span>)</span>),<font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:122_1_1_1_2"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E10:122_1_1_1_2"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E11:122_1_1_1_2"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="simplex0.html#D18" target="_self" title="SIMPLEX0:def.18">Def18</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">Sx</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">Sx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font></span><br/><span class="kw">thus </span><a NAME="E14:122_1_1_1_2"/>
<font color="Maroon" title="c7">Sx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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