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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:116_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="numbers.html#K5" title="NUMBERS:func.5">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="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:116_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="numbers.html#K5" title="NUMBERS:func.5">0</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">  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="simplex0.html#K9" title="SIMPLEX0:func.9">-</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="E2:116_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="numbers.html#K5" title="NUMBERS:func.5">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" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">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="simplex0.html#K9" title="SIMPLEX0:func.9">-</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="simplex0.html#K9" title="SIMPLEX0:func.9">-</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" onmouseover="rs('xxreal_0/D10')" onmouseout="rh()">XXREAL_0:def 10</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">F</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="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="E4:116_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>
 ; <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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:116_1_1_1_2"/><span class="lab"><font color="Green" title="E45">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="numbers.html#K5" title="NUMBERS:func.5">0</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">  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="simplex0.html#K9" title="SIMPLEX0:func.9">-</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="E2:116_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A4</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">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="#E1:116"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="simplex0.html#D18" target="_self" onmouseover="rs('simplex0/D18')" onmouseout="rh()">Def18</a></span>;<br/><a NAME="E3:116_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="#E1:116_1_1_1_2"><span class="lab"><font color="Green" title="E45">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">object</a> <span class="kw"> such that </span><br/><a NAME="E5:116_1_1_1_2"/><span class="lab"><font color="Green" title="E47">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>
 ;<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#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E1:116_1_1_1_2"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E7:116_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="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="xxreal_3.html#T37" onmouseover="rs('xxreal_3/T37')" onmouseout="rh()">XXREAL_3:37</a></span>;<br/><a NAME="E8:116_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">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" onmouseover="rs('xxreal_3/T4')" onmouseout="rh()">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="E10:116_1_1_1_2"/><span class="lab"><font color="Green" title="E49">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="#E2:116_1_1_1_2"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E5:116_1_1_1_2"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="stirl2_1.html#T55" onmouseover="rs('stirl2_1/T55')" onmouseout="rh()">STIRL2_1:55</a></span>;<br/><a NAME="E11:116_1_1_1_2"/><span class="lab"><font color="Green" title="E50">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="numbers.html#K5" title="NUMBERS:func.5">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="#E1:116_1_1_1_2"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E12:116_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="simplex0.html#K9" title="SIMPLEX0:func.9">-</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" onmouseover="rs('xxreal_0/D10')" onmouseout="rh()">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="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1</span>)</span>),<font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:116_1_1_1_2"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E10:116_1_1_1_2"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E11:116_1_1_1_2"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="simplex0.html#D18" target="_self" onmouseover="rs('simplex0/D18')" onmouseout="rh()">Def18</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">Sx</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="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="E14:116_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" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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