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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><br/>  for <font color="Olive" title="b3">s</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2;<br/>
<span class="kw">let </span><font color="Maroon" title="c1">eps</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> implies  ex <font color="Olive" title="b1">F</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">n</font> ) &amp;  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Olive" title="b1">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:5"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</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">  ex <font color="Olive" title="b1">F</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">n</font> ) &amp;  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Olive" title="b1">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">x0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:5"/><span class="lab"><font color="Green" title="E1">A1</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x0</font>
 <span class="kw">and </span><br/><a NAME="E4:5"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
<font color="Maroon" title="c2">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T3" onmouseover="rs('xxreal_3/T3')" onmouseout="rh()">XXREAL_3:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:5"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">and </span><br/><a NAME="E7:5"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="xxreal_3.html#T50" onmouseover="rs('xxreal_3/T50')" onmouseout="rh()">XXREAL_3:50</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E9:5"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>   ex <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>   ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</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" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">x</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">m</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">m</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:5_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">m</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>
<span class="kw">consider </span><font color="Maroon" title="c5">F</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E11:5"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
( <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" onmouseover="rs('recdef_1/S2')" onmouseout="rh()">RECDEF_1:sch 2</a>(<a class="txt" href="#E9:5"><span class="lab"><font color="Green" title="E5">A5</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c5">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"> ( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font> ) &amp;  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> $1;<br/>
<a NAME="E12:5"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_2"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c6">k</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">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:5_2"/>
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:5_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E8">A8</font></span></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>
<a NAME="E13:5"/><span class="lab"><font color="Green" title="E8">A9</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E14:5"/><span class="lab"><font color="Green" title="E9">A10</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</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" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E13:5"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E12:5"><span class="lab"><font color="Green" title="E7">A7</font></span></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">  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font></span><br/>

<a NAME="E15:5"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font> being    <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>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font>
 
;<br/>
<a NAME="E16:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A11</font></span>: 
<font color="Maroon" title="c5">F</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T39" onmouseover="rs('supinf_2/T39')" onmouseout="rh()">SUPINF_2:39</a></span>;<br/>
<a NAME="E17:5"/>
 for <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> $1</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> $1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x0</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">s</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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"> ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> implies <font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_3"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</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"> <font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:5_3"/><span class="lab"><font color="Green" title="E11">A12</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:5_3"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c7">n</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="#E3:5_3"><span class="lab"><font color="Green" title="E11">A12</font></span></a></span>;<br/>
<a NAME="E6:5_3"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">l</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c9">l</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<a NAME="E1:5_3_1"/>
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/>
<a NAME="E2:5_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T105" onmouseover="rs('xxreal_3/T105')" onmouseout="rh()">XXREAL_3:105</a></span>;<br/>
<a NAME="E3:5_3_1"/>
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font> &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="ref" href="supinf_2.html#T40" onmouseover="rs('supinf_2/T40')" onmouseout="rh()">SUPINF_2:40</a></span>;<br/>
<a NAME="E4:5_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A16</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_3_1"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="supinf_2.html#D11" onmouseover="rs('supinf_2/D11')" onmouseout="rh()">SUPINF_2:def 11</a>, <a class="ref" href="xxreal_3.html#T44" onmouseover="rs('xxreal_3/T44')" onmouseout="rh()">XXREAL_3:44</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:5_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x0</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">S<sub>3</sub></font>[<font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">hence </span><a NAME="E6:5_3_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_3_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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>
<a NAME="E7:5_3"/><span class="lab"><font color="Green" title="E14">A17</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="ref" href="supinf_2.html#D11" onmouseover="rs('supinf_2/D11')" onmouseout="rh()">SUPINF_2:def 11</a></span>;<br/>
<a NAME="E8:5_3"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E7:5_3"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="txt" href="#E6:5_3"><span class="lab"><font color="Green" title="E13">A14</font></span></a>);<br/></span>
<a NAME="E9:5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A18</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x0</font>
 
;<br/>
<a NAME="E10:5_3"/>
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">n</font> &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="ref" href="supinf_2.html#T40" onmouseover="rs('supinf_2/T40')" onmouseout="rh()">SUPINF_2:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:5_3"/>
<font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_3"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E9:5_3"><span class="lab"><font color="Green" title="E15">A18</font></span></a>, <a class="ref" href="xxreal_3.html#T48" onmouseover="rs('xxreal_3/T48')" onmouseout="rh()">XXREAL_3:48</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>
<a NAME="E18:5"/><span class="kw">then </span>
<font color="Maroon" title="c2">x0</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of  <a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D1" onmouseover="rs('xxreal_2/D1')" onmouseout="rh()">XXREAL_2:def 1</a></span>;<br/>
<a NAME="E19:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A19</font></span>: 
 <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D3" onmouseover="rs('xxreal_2/D3')" onmouseout="rh()">XXREAL_2:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4"><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="E1:5_4"/>
(  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x0</font> or  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:5"><span class="lab"><font color="Green" title="E11">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1"><span class="kw">suppose </span></a><a NAME="E1:5_4_1"/>
 <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x0</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">  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:5_4_1"/>
 <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_2"><span class="kw">suppose </span></a><a NAME="E1:5_4_2"/>
 <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</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">  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:5_4_2"/>
 <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E2">A2</font></span></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>

</div>
