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<b>let </b><font color="Maroon" title="c1">J</font>, <font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">L</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>; <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">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> ) &amp; <font color="Maroon" title="c1">J</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c2">K</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> ) implies ( <font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">J</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">K</font></span>)</span> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:87"/><i><font color="Green" title="E83">A1</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font>
 <b>and </b><br/><a NAME="E2:87"/><i><font color="Green" title="E84">A2</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font>
 <b>and </b><br/><a NAME="E3:87"/><i><font color="Green" title="E85">A3</font></i>: 
( <font color="Maroon" title="c1">J</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c2">K</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> )
 <b>and </b><br/><a NAME="E4:87"/><i><font color="Green" title="E86">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">J</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">K</font></span>)</span> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E5:87"/><i><font color="Green" title="E87">A5</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green" title="E88">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_1"/>
(  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="#E3:87"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E87"><i><font color="Green" title="E81">Lm9</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_1_1"/><i><font color="Green" title="E89">A7</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> )</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">SJ</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_1_1_1"/>
(  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="#E3:87"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E87"><i><font color="Green" title="E81">Lm9</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_1_1_1_1"/><i><font color="Green" title="E90">A8</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> )</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">SK</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><a NAME="E3:87_1_1_1_1_1"/><i><font color="Green" title="E91">A9</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T7" target="_self" title="MESFUNC5:th.7">Th7</a></i>;<br/><div><i><font color="Green" title="E92">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> implies  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> )</span><br/><b>assume </b><a NAME="E1:87_1_1_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font></span><br/><a NAME="E2:87_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E93">A11</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">SJ'</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:87_1_1_1_1_1_1"/><i><font color="Green" title="E94">A12</font></i>: 
( <font color="Maroon" title="c7">SJ'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">SJ'</font> )
 <b>by </b><i><a class="txt" href="#E1:87_1_1_1"><i><font color="Green" title="E89">A7</font></i></a>, <a class="ref" href="measure6.html#T33" title="MEASURE6:th.33">MEASURE6:33</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">nj</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:87_1_1_1_1_1_1"/><i><font color="Green" title="E95">A13</font></i>: 
( <font color="Maroon" title="c8">nj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">J</font> &amp; <font color="Maroon" title="c7">SJ'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">nj</font> )
 <b>by </b><i><a class="txt" href="#E4:87_1_1_1_1_1_1"><i><font color="Green" title="E94">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">nj</font> = <font color="Maroon" title="c8">nj</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E6:87_1_1_1_1_1_1"><i><font color="Green" title="E95">A13</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">SK'</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E9:87_1_1_1_1_1_1"/><i><font color="Green" title="E96">A14</font></i>: 
( <font color="Maroon" title="c10">SK'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">SK'</font> )
 <b>by </b><i><a class="txt" href="#E1:87_1_1_1_1_1"><i><font color="Green" title="E90">A8</font></i></a>, <a class="txt" href="#E2:87_1_1_1_1_1_1"><i><font color="Green" title="E93">A11</font></i></a>, <a class="ref" href="measure6.html#T33" title="MEASURE6:th.33">MEASURE6:33</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">nk</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:87_1_1_1_1_1_1"/><i><font color="Green" title="E97">A15</font></i>: 
( <font color="Maroon" title="c11">nk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c10">SK'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">nk</font> )
 <b>by </b><i><a class="txt" href="#E9:87_1_1_1_1_1_1"><i><font color="Green" title="E96">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">nk</font> = <font color="Maroon" title="c11">nk</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E11:87_1_1_1_1_1_1"><i><font color="Green" title="E97">A15</font></i></a></i>;<br/><a NAME="E13:87_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 ;<br/><a NAME="E14:87_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E98">A16</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T7" target="_self" title="MESFUNC5:th.7">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">n</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c9">nj</font>,<font color="Maroon" title="c12">nk</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>take </b><font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c13">n</font>; <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">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font></span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span>
<div class="add"><b>let </b><font color="Maroon" title="c15">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> )</span><br/><b>assume </b><a NAME="E1:87_1_1_1_1_1_1_1"/><i><font color="Green" title="E99">A17</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font></span><br/><b>reconsider </b><font color="Maroon" title="c16">m1</font> = <font color="Maroon" title="c15">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:87_1_1_1_1_1_1_1"/>
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E4:87_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E100">A18</font></i>: 
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/><a NAME="E5:87_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="#E5:87"><i><font color="Green" title="E87">A5</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E6:87_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E101">A19</font></i>: 
( <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font> )
 <b>by </b><i><a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></i>;<br/><a NAME="E7:87_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E102">A20</font></i>: 
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="#E3:87"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="#E4:87_1_1_1_1_1_1_1"><i><font color="Green" title="E100">A18</font></i></a>, <a class="ref" href="mesfunc5.html#D5" target="_self" title="MESFUNC5:def.5">Def5</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:87_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E103">A21</font></i>: 
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_2.html#T8" title="SUPINF_2:th.8">SUPINF_2:8</a>, <a class="ref" href="supinf_2.html#T9" title="SUPINF_2:th.9">SUPINF_2:9</a></i>;<br/><a NAME="E9:87_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/><a NAME="E10:87_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span> &amp; <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="#E3:87_1_1_1_1_1"><i><font color="Green" title="E91">A9</font></i></a>, <a class="txt" href="#E6:87_1_1_1_1_1_1_1"><i><font color="Green" title="E101">A19</font></i></a>, <a class="txt" href="#E8:87_1_1_1_1_1_1_1"><i><font color="Green" title="E103">A21</font></i></a>, <a class="ref" href="supinf_2.html#T14" title="SUPINF_2:th.14">SUPINF_2:14</a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><a NAME="E11:87_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E104">A22</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T72" title="SUPINF_2:th.72">SUPINF_2:72</a></i>;<br/><a NAME="E12:87_1_1_1_1_1_1_1"/><i><font color="Green" title="E105">A23</font></i>: 
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T15" title="EXTREAL2:th.15">EXTREAL2:15</a></i>;<br/><a NAME="E13:87_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E11:87_1_1_1_1_1_1_1"><i><font color="Green" title="E104">A22</font></i></a>, <a class="ref" href="extreal1.html#T25" title="EXTREAL1:th.25">EXTREAL1:25</a></i>;<br/><a NAME="E14:87_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E106">A24</font></i>: 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:87_1_1_1_1_1_1_1"><i><font color="Green" title="E105">A23</font></i></a>, <a class="ref" href="extreal2.html#T55" title="EXTREAL2:th.55">EXTREAL2:55</a></i>;<br/><i><font color="Green" title="E107">A25</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:87_1_1_1_1_1_1_1"><i><font color="Green" title="E102">A20</font></i></a>, <a class="ref" href="extreal1.html#T11" title="EXTREAL1:th.11">EXTREAL1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
<b>by </b><i><a class="ref" href="extreal1.html#T11" title="EXTREAL1:th.11">EXTREAL1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">SK</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">SJ</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:87_1_1_1_1_1"><i><font color="Green" title="E91">A9</font></i></a>, <a class="txt" href="#E4:87_1_1_1_1_1_1_1"><i><font color="Green" title="E100">A18</font></i></a>, <a class="txt" href="#E6:87_1_1_1_1_1_1_1"><i><font color="Green" title="E101">A19</font></i></a>, <a class="ref" href="extreal2.html#T40" title="EXTREAL2:th.40">EXTREAL2:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">SK</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">SJ</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E86">A4</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c17">SJ'</font> = <font color="Maroon" title="c7">SJ'</font>, <font color="Maroon" title="c18">SK'</font> = <font color="Maroon" title="c10">SK'</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E17:87_1_1_1_1_1_1_1"/>
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c17">SJ'</font> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c18">SK'</font> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="#E4:87_1_1_1_1_1_1"><i><font color="Green" title="E94">A12</font></i></a>, <a class="txt" href="#E9:87_1_1_1_1_1_1"><i><font color="Green" title="E96">A14</font></i></a>, <a class="ref" href="extreal2.html#T29" title="EXTREAL2:th.29">EXTREAL2:29</a></i>;<br/><a NAME="E18:87_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E108">A26</font></i>: 
( <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">SJ'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">SK'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="extreal2.html#T30" title="EXTREAL2:th.30">EXTREAL2:30</a></i>;<br/><a NAME="E19:87_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">nk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E20:87_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">nk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 <b>by </b><i><a class="txt" href="#E1:87_1_1_1_1_1_1_1"><i><font color="Green" title="E99">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E21:87_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">nk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font>
 <b>by </b><i><a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/><a NAME="E22:87_1_1_1_1_1_1_1"/><b>then </b>
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">SK'</font> &amp; <font color="Maroon" title="c18">SK'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="#E3:87"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="#E11:87_1_1_1_1_1_1"><i><font color="Green" title="E97">A15</font></i></a>, <a class="txt" href="#E7:87_1_1_1_1_1_1_1"><i><font color="Green" title="E102">A20</font></i></a>, <a class="ref" href="mesfunc5.html#D5" target="_self" title="MESFUNC5:def.5">Def5</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">s0</font> = <font color="Maroon" title="c18">SK'</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">s1</font> =  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E25:87_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">SK'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c19">s0</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><a NAME="E26:87_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E109">A27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">SJ'</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">SK'</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">SK'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E18:87_1_1_1_1_1_1_1"><i><font color="Green" title="E108">A26</font></i></a>, <a class="ref" href="measure6.html#T12" title="MEASURE6:th.12">MEASURE6:12</a></i>;<br/><a NAME="E27:87_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">SK'</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E18:87_1_1_1_1_1_1_1"><i><font color="Green" title="E108">A26</font></i></a>, <a class="ref" href="measure6.html#T12" title="MEASURE6:th.12">MEASURE6:12</a></i>;<br/><a NAME="E28:87_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E110">A28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">SJ'</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">SK'</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E26:87_1_1_1_1_1_1_1"><i><font color="Green" title="E109">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E29:87_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c9">nj</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font> &amp; <font color="Maroon" title="c12">nk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E30:87_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">nj</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c12">nk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> )
 <b>by </b><i><a class="txt" href="#E1:87_1_1_1_1_1_1_1"><i><font color="Green" title="E99">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E31:87_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c17">SJ'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c18">SK'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> )
 <b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="#E6:87_1_1_1_1_1_1"><i><font color="Green" title="E95">A13</font></i></a>, <a class="txt" href="#E11:87_1_1_1_1_1_1"><i><font color="Green" title="E97">A15</font></i></a></i>;<br/><a NAME="E32:87_1_1_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">SJ'</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">SK'</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/><a NAME="E33:87_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></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="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">SJ</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c17">SJ'</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">SK</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c18">SK'</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T14" title="SUPINF_2:th.14">SUPINF_2:14</a></i>;<br/><b>hence </b><a NAME="E34:87_1_1_1_1_1_1_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">SJ</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">SK</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="#E14:87_1_1_1_1_1_1"><i><font color="Green" title="E98">A16</font></i></a>, <a class="txt" href="#E14:87_1_1_1_1_1_1_1"><i><font color="Green" title="E106">A24</font></i></a>, <a class="txt" href="#E15:87_1_1_1_1_1_1_1"><i><font color="Green" title="E107">A25</font></i></a>, <a class="txt" href="#E28:87_1_1_1_1_1_1_1"><i><font color="Green" title="E110">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div></div><b>end;</b></div><a NAME="E5:87_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E93">A30</font></i>: 
<font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D8" target="_self" title="MESFUNC5:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E6:87_1_1_1_1_1"/>
<font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span><br/><b>hence </b><a NAME="E7:87_1_1_1_1_1"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:87_1_1_1_1_1"><i><font color="Green" title="E91">A9</font></i></a>, <a class="txt" href="#E4:87_1_1_1_1_1"><i><font color="Green" title="E92">A10</font></i></a>, <a class="txt" href="#E5:87_1_1_1_1_1"><i><font color="Green" title="E93">A30</font></i></a>, <a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_1_1_1_2"/><i><font color="Green" title="E90">A31</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> )</span><br/><div class="add"><a NAME="E2:87_1_1_1_1_2"/>
for <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font> implies  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> )</span><br/>

<b>assume </b><a NAME="E1:87_1_1_1_1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font></span><br/>

<a NAME="E2:87_1_1_1_1_2_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">j</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:87_1_1_1_1_2_1"/><i><font color="Green" title="E91">A32</font></i>: 
( <font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> )
 <b>by </b><i><a class="txt" href="#E1:87_1_1_1"><i><font color="Green" title="E89">A7</font></i></a>, <a class="ref" href="measure6.html#T33" title="MEASURE6:th.33">MEASURE6:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">n1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:87_1_1_1_1_2_1"/><i><font color="Green" title="E92">A33</font></i>: 
( <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">J</font> &amp; <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">n1</font> )
 <b>by </b><i><a class="txt" href="#E4:87_1_1_1_1_2_1"><i><font color="Green" title="E91">A32</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">n1</font> = <font color="Maroon" title="c7">n1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E6:87_1_1_1_1_2_1"><i><font color="Green" title="E92">A33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">supj</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E1:87_1_1_1"><i><font color="Green" title="E89">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">p'2</font> = <font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2, <font color="Maroon" title="c11">p'</font> = <font color="Maroon" title="c5">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E10:87_1_1_1_1_2_1"/><i><font color="Green" title="E93">A34</font></i>: 
<font color="Maroon" title="c9">supj</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">p'2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E11:87_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E94">A35</font></i>: 
<font color="Maroon" title="c11">p'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">supj</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">p'2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E12:87_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:87_1_1_1_1_2"><i><font color="Green" title="E90">A31</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">k</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E14:87_1_1_1_1_2_1"/><i><font color="Green" title="E95">A36</font></i>: 
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</font> )
 <b>by </b><i><a class="ref" href="hahnban.html#T8" title="HAHNBAN:th.8">HAHNBAN:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">n2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:87_1_1_1_1_2_1"/><i><font color="Green" title="E96">A37</font></i>: 
( <font color="Maroon" title="c13">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">n2</font> )
 <b>by </b><i><a class="txt" href="#E14:87_1_1_1_1_2_1"><i><font color="Green" title="E95">A36</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">n2</font> = <font color="Maroon" title="c13">n2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E16:87_1_1_1_1_2_1"><i><font color="Green" title="E96">A37</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">n</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font>;<br/>
<a NAME="E18:87_1_1_1_1_2_1"/>
( <font color="Maroon" title="c8">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font> &amp; <font color="Maroon" title="c14">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E19:87_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/>
<a NAME="E20:87_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E97">A38</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E4:87_1_1_1_1_2_1"><i><font color="Green" title="E91">A32</font></i></a>, <a class="txt" href="#E6:87_1_1_1_1_2_1"><i><font color="Green" title="E92">A33</font></i></a>, <a class="txt" href="#E14:87_1_1_1_1_2_1"><i><font color="Green" title="E95">A36</font></i></a>, <a class="txt" href="#E16:87_1_1_1_1_2_1"><i><font color="Green" title="E96">A37</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E21:87_1_1_1_1_2_1"/><i><font color="Green" title="E98">A39</font></i>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E11:87_1_1_1_1_2_1"><i><font color="Green" title="E94">A35</font></i></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/>
<a NAME="E22:87_1_1_1_1_2_1"/>
<font color="Maroon" title="c11">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">supj</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">p'2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">supj</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">p'2</font></span>)</span>
 
;<br/>
<a NAME="E23:87_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p'</font>
 
<b>by </b><i><a class="txt" href="#E10:87_1_1_1_1_2_1"><i><font color="Green" title="E93">A34</font></i></a>, <a class="txt" href="#E11:87_1_1_1_1_2_1"><i><font color="Green" title="E94">A35</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/>
<a NAME="E24:87_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E99">A43</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E20:87_1_1_1_1_2_1"><i><font color="Green" title="E97">A38</font></i></a>, <a class="txt" href="#E21:87_1_1_1_1_2_1"><i><font color="Green" title="E98">A39</font></i></a>, <a class="ref" href="mesfunc5.html#T10" target="_self" title="MESFUNC5:th.10">Th10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">m</font> implies <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m</font> )</span><br/><b>assume </b><a NAME="E1:87_1_1_1_1_2_1_1"/>
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m</font></span><br/><a NAME="E2:87_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E100">A44</font></i>: 
( <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m</font> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m</font> )
 <b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">m1</font> = <font color="Maroon" title="c16">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:87_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:87_1_1_1_1_2_1_1"><i><font color="Green" title="E100">A44</font></i></a>, <a class="ref" href="supinf_2.html#T14" title="SUPINF_2:th.14">SUPINF_2:14</a></i>;<br/><a NAME="E5:87_1_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c14">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m</font>
 <b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E6:87_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">m</font>
 <b>by </b><i><a class="txt" href="#E24:87_1_1_1_1_2_1"><i><font color="Green" title="E99">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E26:87_1_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:87_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E91">A45</font></i>: 
<font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" target="_self" title="MESFUNC5:def.9">Def9</a></i>;<br/><b>hence </b><a NAME="E4:87_1_1_1_1_2"/>
<font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span><br/><a NAME="E5:87_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E92">A46</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E3:87_1_1_1_1_2"><i><font color="Green" title="E91">A45</font></i></a>, <a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></i>;<br/><a NAME="E6:87_1_1_1_1_2"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:87_1_1_1"><i><font color="Green" title="E89">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/><b>hence </b><a NAME="E7:87_1_1_1_1_2"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:87_1_1_1_1_2"><i><font color="Green" title="E90">A31</font></i></a>, <a class="txt" href="#E5:87_1_1_1_1_2"><i><font color="Green" title="E92">A46</font></i></a>, <a class="ref" href="supinf_2.html#D2" title="SUPINF_2:def.2">SUPINF_2:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_1_2"/><i><font color="Green" title="E89">A47</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font> implies  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive">b<sub>3</sub></font> )</span><br/><b>assume </b><a NAME="E1:87_1_1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive">b<sub>3</sub></font></span><br/><a NAME="E2:87_1_1_2_1"/><b>then </b><i><font color="Green" title="E90">A48</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_1_2_1_1"/>
(  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="#E3:87"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E87"><i><font color="Green" title="E81">Lm9</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_1_2_1_1_1"/><i><font color="Green" title="E91">A49</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">k</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:87_1_1_2_1_1_1"/><i><font color="Green" title="E92">A50</font></i>: 
( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> )
 <b>by </b><i><a class="txt" href="#E2:87_1_1_2_1"><i><font color="Green" title="E90">A48</font></i></a>, <a class="ref" href="measure6.html#T33" title="MEASURE6:th.33">MEASURE6:33</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">n1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:87_1_1_2_1_1_1"/><i><font color="Green" title="E93">A51</font></i>: 
( <font color="Maroon" title="c6">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">n1</font> )
 <b>by </b><i><a class="txt" href="#E3:87_1_1_2_1_1_1"><i><font color="Green" title="E92">A50</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">n1</font> = <font color="Maroon" title="c6">n1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E5:87_1_1_2_1_1_1"><i><font color="Green" title="E93">A51</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">supk</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E1:87_1_1_2_1_1_1"><i><font color="Green" title="E91">A49</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">p'2</font> = <font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2, <font color="Maroon" title="c10">p'</font> = <font color="Maroon" title="c4">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E9:87_1_1_2_1_1_1"/><i><font color="Green" title="E94">A52</font></i>: 
<font color="Maroon" title="c8">supk</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">p'2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><a NAME="E10:87_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E95">A53</font></i>: 
<font color="Maroon" title="c10">p'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">supk</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">p'2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><a NAME="E11:87_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:87_1_1_2"><i><font color="Green" title="E89">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">j</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E13:87_1_1_2_1_1_1"/><i><font color="Green" title="E96">A54</font></i>: 
( <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">j</font> )
 <b>by </b><i><a class="ref" href="hahnban.html#T8" title="HAHNBAN:th.8">HAHNBAN:8</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">n2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:87_1_1_2_1_1_1"/><i><font color="Green" title="E97">A55</font></i>: 
( <font color="Maroon" title="c12">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">J</font> &amp; <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">n2</font> )
 <b>by </b><i><a class="txt" href="#E13:87_1_1_2_1_1_1"><i><font color="Green" title="E96">A54</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">n2</font> = <font color="Maroon" title="c12">n2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E15:87_1_1_2_1_1_1"><i><font color="Green" title="E97">A55</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c14">n</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font>;<br/><a NAME="E17:87_1_1_2_1_1_1"/>
( <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font> &amp; <font color="Maroon" title="c13">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E18:87_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/><a NAME="E19:87_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E98">A56</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:87_1_1_2_1_1_1"><i><font color="Green" title="E92">A50</font></i></a>, <a class="txt" href="#E5:87_1_1_2_1_1_1"><i><font color="Green" title="E93">A51</font></i></a>, <a class="txt" href="#E13:87_1_1_2_1_1_1"><i><font color="Green" title="E96">A54</font></i></a>, <a class="txt" href="#E15:87_1_1_2_1_1_1"><i><font color="Green" title="E97">A55</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E20:87_1_1_2_1_1_1"/><i><font color="Green" title="E99">A57</font></i>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:87_1_1_2_1_1_1"><i><font color="Green" title="E95">A53</font></i></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/><a NAME="E21:87_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">supk</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">p'2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">supk</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">p'2</font></span>)</span>
 ;<br/><a NAME="E22:87_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p'</font>
 <b>by </b><i><a class="txt" href="#E9:87_1_1_2_1_1_1"><i><font color="Green" title="E94">A52</font></i></a>, <a class="txt" href="#E10:87_1_1_2_1_1_1"><i><font color="Green" title="E95">A53</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/><a NAME="E23:87_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E100">A61</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E19:87_1_1_2_1_1_1"><i><font color="Green" title="E98">A56</font></i></a>, <a class="txt" href="#E20:87_1_1_2_1_1_1"><i><font color="Green" title="E99">A57</font></i></a>, <a class="ref" href="mesfunc5.html#T10" target="_self" title="MESFUNC5:th.10">Th10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> implies <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> )</span><br/><b>assume </b><a NAME="E1:87_1_1_2_1_1_1_1"/>
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span><br/><a NAME="E2:87_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E101">A62</font></i>: 
( <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font> )
 <b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">m1</font> = <font color="Maroon" title="c15">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:87_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:87_1_1_2_1_1_1_1"><i><font color="Green" title="E101">A62</font></i></a>, <a class="ref" href="supinf_2.html#T14" title="SUPINF_2:th.14">SUPINF_2:14</a></i>;<br/><a NAME="E5:87_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c13">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font>
 <b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E6:87_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">m</font>
 <b>by </b><i><a class="txt" href="#E23:87_1_1_2_1_1_1"><i><font color="Green" title="E100">A61</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E25:87_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_1_2_1_1_2"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:87_1_1_2_1_1_2"/><i><font color="Green" title="E91">A63</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n1</font>
 <b>by </b><i><a class="txt" href="#E3:87"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="#E2:87_1_1_2_1"><i><font color="Green" title="E90">A48</font></i></a>, <a class="ref" href="mesfunc5.html#T65" target="_self" title="MESFUNC5:th.65">Th65</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">n2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:87_1_1_2_1_1_2"/><i><font color="Green" title="E92">A64</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">n2</font>
 <b>by </b><i><a class="txt" href="#E3:87"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="#E1:87_1_1_2"><i><font color="Green" title="E89">A47</font></i></a>, <a class="txt" href="#E2:87_1_1_2_1"><i><font color="Green" title="E90">A48</font></i></a>, <a class="ref" href="mesfunc5.html#T65" target="_self" title="MESFUNC5:th.65">Th65</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">n1</font> = <font color="Maroon" title="c5">n1</font>, <font color="Maroon" title="c8">n2</font> = <font color="Maroon" title="c6">n2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon" title="c9">n</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font>;<br/><a NAME="E7:87_1_1_2_1_1_2"/>
( <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font> &amp; <font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E8:87_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/><a NAME="E9:87_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E93">A65</font></i>: 
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:87_1_1_2_1_1_2"><i><font color="Green" title="E91">A63</font></i></a>, <a class="txt" href="#E5:87_1_1_2_1_1_2"><i><font color="Green" title="E92">A64</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:87_1_1_2_1_1_2"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/><a NAME="E11:87_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E94">A66</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:87_1_1_2_1_1_2"><i><font color="Green" title="E93">A65</font></i></a>, <a class="ref" href="mesfunc5.html#T10" target="_self" title="MESFUNC5:th.10">Th10</a></i>;<br/><a NAME="E12:87_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 ;<br/><a NAME="E13:87_1_1_2_1_1_2"/><b>then </b>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:87_1_1_2_1_1_2"><i><font color="Green" title="E94">A66</font></i></a>, <a class="ref" href="mesfunc5.html#T7" target="_self" title="MESFUNC5:th.7">Th7</a></i>;<br/><a NAME="E14:87_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E95">A67</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> implies <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">m</font> )</span><br/><b>assume </b><a NAME="E1:87_1_1_2_1_1_2_1"/>
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">m</font></span><br/><a NAME="E2:87_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E96">A68</font></i>: 
( <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">m</font> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">m</font> )
 <b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/><a NAME="E3:87_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:87_1_1_2_1_1_2_1"><i><font color="Green" title="E96">A68</font></i></a>, <a class="ref" href="supinf_2.html#T14" title="SUPINF_2:th.14">SUPINF_2:14</a></i>;<br/><a NAME="E4:87_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/><a NAME="E5:87_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c8">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E6:87_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="#E14:87_1_1_2_1_1_2"><i><font color="Green" title="E95">A67</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E16:87_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:87_1_1_2"/><b>then </b><i><font color="Green" title="E90">A69</font></i>: 
<font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" target="_self" title="MESFUNC5:def.9">Def9</a></i>;<br/><b>hence </b><a NAME="E4:87_1_1_2"/>
<font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span><br/><a NAME="E5:87_1_1_2"/><b>then </b><i><font color="Green" title="E91">A70</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E3:87_1_1_2"><i><font color="Green" title="E90">A69</font></i></a>, <a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></i>;<br/><a NAME="E6:87_1_1_2"/>
<font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="#E5:87"><i><font color="Green" title="E87">A5</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E7:87_1_1_2"/><b>then </b>
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:87"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="mesfunc5.html#D5" target="_self" title="MESFUNC5:def.5">Def5</a>, <a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></i>;<br/><b>hence </b><a NAME="E8:87_1_1_2"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:87_1_1_2"><i><font color="Green" title="E89">A47</font></i></a>, <a class="txt" href="#E5:87_1_1_2"><i><font color="Green" title="E91">A70</font></i></a>, <a class="ref" href="supinf_2.html#D2" title="SUPINF_2:def.2">SUPINF_2:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:87"/>
<font color="Maroon" title="c3">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">J</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">K</font></span>)</span> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> )</span><br/>

<div><i><font color="Green" title="E89">A71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> implies <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> )</span><br/><b>assume </b><a NAME="E1:87_2"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font></span><br/><a NAME="E2:87_2"/><b>then </b><i><font color="Green" title="E90">A72</font></i>: 
( <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> )
 <b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/><a NAME="E3:87_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:87_2"><i><font color="Green" title="E90">A72</font></i></a>, <a class="ref" href="supinf_2.html#T14" title="SUPINF_2:th.14">SUPINF_2:14</a></i>;<br/><a NAME="E4:87_2"/><b>then </b>
<font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E5:87_2"/>
<font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E86">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:87"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c3">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T60" target="_self" title="MESFUNC5:th.60">Th60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">J</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">K</font></span>)</span> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E10:87"/>
(  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="#E2:87"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="mesfunc5.html#T60" target="_self" title="MESFUNC5:th.60">Th60</a></i>;<br/>
<b>hence </b><a NAME="E11:87"/>
(  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">J</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">K</font></span>)</span> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E6:87"><i><font color="Green" title="E88">A6</font></i></a>, <a class="txt" href="#E8:87"><i><font color="Green" title="E89">A71</font></i></a>, <a class="ref" href="mesfunc5.html#T60" target="_self" title="MESFUNC5:th.60">Th60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
