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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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#K17" title="SUPINF_2:func.17">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#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:81"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
 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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font>
 <span class="kw">and </span><br/><a NAME="E2:81"/><span class="lab"><font color="Green" title="E78">A2</font></span>: 
 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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font>
 <span class="kw">and </span><br/><a NAME="E3:81"/><span class="lab"><font color="Green" title="E79">A3</font></span>: 
<font color="Maroon" title="c1">J</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> 
 <span class="kw">and </span><br/><a NAME="E4:81"/><span class="lab"><font color="Green" title="E80">A4</font></span>: 
<font color="Maroon" title="c2">K</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> 
 <span class="kw">and </span><br/><a NAME="E5:81"/><span class="lab"><font color="Green" title="E81">A5</font></span>: 
 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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K17" title="SUPINF_2:func.17">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#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E6:81"/><span class="lab"><font color="Green" title="E82">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:81"/><span class="lab"><font color="Green" title="E83">A7</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E84">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:81_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#K17" title="SUPINF_2:func.17">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#K17" title="SUPINF_2:func.17">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="txt" href="mesfunc5.html#E80"><span class="lab"><font color="Green" title="E75">Lm8</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:81_1_1_1"/><span class="lab"><font color="Green" title="E85">A9</font></span>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K17" title="SUPINF_2:func.17">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="81_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:81_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#K17" title="SUPINF_2:func.17">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#K17" title="SUPINF_2:func.17">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="txt" href="mesfunc5.html#E80"><span class="lab"><font color="Green" title="E75">Lm8</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:81_1_1_1_1_1"/><span class="lab"><font color="Green" title="E86">A10</font></span>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K17" title="SUPINF_2:func.17">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/><div><span class="lab"><font color="Green" title="E87">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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/><span class="kw">assume </span><a NAME="E1:81_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E88">A12</font></span>: 
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">SJ9</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> <span class="kw"> such that </span><br/><a NAME="E3:81_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E89">A13</font></span>: 
<font color="Maroon" title="c7">SJ9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font>
 <span class="kw">and </span><br/><a NAME="E4:81_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E90">A14</font></span>: 
<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#K17" title="SUPINF_2:func.17">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">SJ9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1"><span class="lab"><font color="Green" title="E85">A9</font></span></a>, <a class="ref" href="measure6.html#T6" title="MEASURE6:th.6">MEASURE6:6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">nj</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:81_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E91">A15</font></span>: 
<font color="Maroon" title="c8">nj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">J</font>
 <span class="kw">and </span><br/><a NAME="E7:81_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E92">A16</font></span>: 
<font color="Maroon" title="c7">SJ9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E89">A13</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E91">A15</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">SK9</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> <span class="kw"> such that </span><br/><a NAME="E10:81_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E93">A17</font></span>: 
<font color="Maroon" title="c10">SK9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E11:81_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E94">A18</font></span>: 
<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#K17" title="SUPINF_2:func.17">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">SK9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A10</font></span></a>, <a class="txt" href="#E1:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E88">A12</font></span></a>, <a class="ref" href="measure6.html#T6" title="MEASURE6:th.6">MEASURE6:6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">nk</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:81_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E95">A19</font></span>: 
<font color="Maroon" title="c11">nk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E14:81_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E96">A20</font></span>: 
<font color="Maroon" title="c10">SK9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E93">A17</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E95">A19</font></span></a></span>;<br/><span class="kw">reconsider </span><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/><span class="kw">take </span><font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c13">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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="E17:81_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="E18:81_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A21</font></span>: 
 <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#K3" title="SUPINF_2:func.3">+</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c15">SJ9</font> = <font color="Maroon" title="c7">SJ9</font>, <font color="Maroon" title="c16">SK9</font> = <font color="Maroon" title="c10">SK9</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c17">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c17">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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/><span class="kw">assume </span><a NAME="E2:81_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E98">A22</font></span>: 
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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="E3:81_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E4:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">nk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E98">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">SK9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E14:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E96">A20</font></span></a></span>;<br/><a NAME="E6:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A23</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="c16">SK9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T37" title="XXREAL_3:th.37">XXREAL_3:37</a></span>;<br/><a NAME="E7:81_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E8:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">nj</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E98">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">SJ9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E7:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E92">A16</font></span></a></span>;<br/><a NAME="E10:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="c15">SJ9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T37" title="XXREAL_3:th.37">XXREAL_3:37</a></span>;<br/><a NAME="E11:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A24</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="c15">SJ9</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c16">SK9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E99">A23</font></span></a>, <a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/><a NAME="E12:81_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E13:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A25</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">m1</font> = <font color="Maroon" title="c17">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E16:81_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E102">A26</font></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"><font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T26" title="XXREAL_3:th.26">XXREAL_3:26</a></span>;<br/><a NAME="E17:81_1_1_1_1_1_1_1"/>
 <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#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c16">SK9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E94">A18</font></span></a>, <a class="ref" href="xxreal_3.html#T54" title="XXREAL_3:th.54">XXREAL_3:54</a></span>;<br/><a NAME="E18:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<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="c16">SK9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T55" title="XXREAL_3:th.55">XXREAL_3:55</a></span>;<br/><a NAME="E19:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A27</font></span>: 
<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#K3" title="SUPINF_2:func.3">+</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="c16">SK9</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#K3" title="SUPINF_2:func.3">+</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T43" title="XXREAL_3:th.43">XXREAL_3:43</a></span>;<br/><a NAME="E20:81_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#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c15">SJ9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E90">A14</font></span></a>, <a class="ref" href="xxreal_3.html#T54" title="XXREAL_3:th.54">XXREAL_3:54</a></span>;<br/><a NAME="E21:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A28</font></span>: 
<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="c15">SJ9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T55" title="XXREAL_3:th.55">XXREAL_3:55</a></span>;<br/><a NAME="E22:81_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E23:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">nk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E98">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E24:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A29</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a></span>;<br/><a NAME="E25:81_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E106">A30</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E26:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A31</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/><a NAME="E27:81_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E82">A6</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><a NAME="E28:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A32</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></span>;<br/><a NAME="E29:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A33</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E106">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/><a NAME="E30:81_1_1_1_1_1_1_1"/>
 <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="c16">SK9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="txt" href="#E14:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E96">A20</font></span></a>, <a class="ref" href="mesfunc5.html#D5" target="_self" title="MESFUNC5:def.5">Def5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">s0</font> = <font color="Maroon" title="c16">SK9</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E96">A20</font></span></a>, <a class="txt" href="#E29:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E109">A33</font></span></a>, <a class="txt" href="#E24:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E105">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/><a NAME="E32:81_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E110">A34</font></span>: 
<font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E81">A5</font></span></a></span>;<br/><a NAME="E33:81_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81"><span class="lab"><font color="Green" title="E83">A7</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><a NAME="E34:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A35</font></span>: 
<font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></span>;<br/><a NAME="E35:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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#K3" title="SUPINF_2:func.3">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E108">A32</font></span></a>, <a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/><a NAME="E36:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E110">A34</font></span></a>, <a class="txt" href="#E16:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A26</font></span></a>, <a class="ref" href="xxreal_3.html#T40" title="XXREAL_3:th.40">XXREAL_3:40</a></span>;<br/><a NAME="E37:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A36</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A26</font></span></a>, <a class="ref" href="extreal2.html#T7" title="EXTREAL2:th.7">EXTREAL2:7</a></span>;<br/><a NAME="E38:81_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="c16">SK9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c20">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></span>;<br/><a NAME="E39:81_1_1_1_1_1_1_1"/><span class="kw">then </span>
<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="c15">SJ9</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c16">SK9</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#K3" title="SUPINF_2:func.3">+</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="c16">SK9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A28</font></span></a>, <a class="ref" href="xxreal_3.html#T43" title="XXREAL_3:th.43">XXREAL_3:43</a></span>;<br/><a NAME="E40:81_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A37</font></span>: 
<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="c15">SJ9</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c16">SK9</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#K3" title="SUPINF_2:func.3">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E103">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E41:81_1_1_1_1_1_1_1"/>
 <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="ref" href="mesfunc5.html#D5" target="_self" title="MESFUNC5:def.5">Def5</a></span>;<br/><span class="kw">then </span><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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E109">A33</font></span></a>, <a class="ref" href="xxreal_3.html#T30" title="XXREAL_3:th.30">XXREAL_3:30</a></span>
<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#K3" title="SUPINF_2:func.3">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T30" title="XXREAL_3:th.30">XXREAL_3:30</a></span>
<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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E101">A25</font></span></a>, <a class="txt" href="#E26:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E107">A31</font></span></a>, <a class="txt" href="#E34:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E111">A35</font></span></a>, <a class="txt" href="#E28:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E108">A32</font></span></a>, <a class="ref" href="xxreal_3.html#T31" title="XXREAL_3:th.31">XXREAL_3:31</a></span>
<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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E81">A5</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E43:81_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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">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="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E18:81_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E97">A21</font></span></a>, <a class="txt" href="#E37:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E112">A36</font></span></a>, <a class="txt" href="#E40:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E113">A37</font></span></a>, <a class="txt" href="#E11:81_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E4:81_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A38</font></span>: 
<font color="Maroon" title="c3">L</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D8" target="_self" title="MESFUNC5:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E5:81_1_1_1_1_1"/>
<font color="Maroon" title="c3">L</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span><br/><span class="kw">hence </span><a NAME="E6:81_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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1_1_1_1"><span class="lab"><font color="Green" title="E87">A11</font></span></a>, <a class="txt" href="#E4:81_1_1_1_1_1"><span class="lab"><font color="Green" title="E88">A38</font></span></a>, <a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:81_1_1_1_1_2"/><span class="lab"><font color="Green" title="E86">A39</font></span>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> )</span><br/><div class="add"><a NAME="E2:81_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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">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#K17" title="SUPINF_2:func.17">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1"><span class="lab"><font color="Green" title="E85">A9</font></span></a></span>;<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">p92</font> = <font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2, <font color="Maroon" title="c8">p9</font> = <font color="Maroon" title="c6">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:81_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="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" 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="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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> <span class="kw"> such that </span><br/><a NAME="E5:81_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E87">A40</font></span>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font>
 <span class="kw">and </span><br/><a NAME="E6:81_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E88">A41</font></span>: 
<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#K17" title="SUPINF_2:func.17">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="c9">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1"><span class="lab"><font color="Green" title="E85">A9</font></span></a>, <a class="ref" href="measure6.html#T6" title="MEASURE6:th.6">MEASURE6:6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">n1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:81_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E89">A42</font></span>: 
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">J</font>
 <span class="kw">and </span><br/><a NAME="E9:81_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E90">A43</font></span>: 
<font color="Maroon" title="c9">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="c10">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E87">A40</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E10:81_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E91">A44</font></span>: 
<font color="Maroon" title="c5">supj</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">p92</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#K17" title="SUPINF_2:func.17">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></span>;<br/>
<a NAME="E11:81_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A45</font></span>: 
<font color="Maroon" title="c8">p9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">supj</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">p92</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="c6">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#K17" title="SUPINF_2:func.17">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="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></span>;<br/>
<a NAME="E12:81_1_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">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#K17" title="SUPINF_2:func.17">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="c6">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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">k</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><span class="kw"> such that </span><br/><a NAME="E14:81_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E93">A46</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E15:81_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E94">A47</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">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#K17" title="SUPINF_2:func.17">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="c6">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">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T94" title="XXREAL_2:th.94">XXREAL_2:94</a></span>;<br/>
<a NAME="E16:81_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">supj</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">p92</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="c5">supj</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">p92</font></span>)</span>
 
;<br/>
<a NAME="E17:81_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A48</font></span>: 
<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="c6">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#K17" title="SUPINF_2:func.17">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="c6">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#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">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="c6">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="c8">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E91">A44</font></span></a>, <a class="txt" href="#E11:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E92">A45</font></span></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">n1</font> = <font color="Maroon" title="c10">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E89">A42</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">n2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E20:81_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E96">A49</font></span>: 
<font color="Maroon" title="c13">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E21:81_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E97">A50</font></span>: 
<font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E93">A46</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E96">A49</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">n</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c14">n2</font>);<br/>
<a NAME="E23:81_1_1_1_1_2_1"/>
<font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c14">n2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E24:81_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A51</font></span>: 
<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#K17" title="SUPINF_2:func.17">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="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c14">n2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E88">A41</font></span></a>, <a class="txt" href="#E9:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E90">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E25:81_1_1_1_1_2_1"/>
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c14">n2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E26:81_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A52</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">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#K17" title="SUPINF_2:func.17">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="c6">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c14">n2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E94">A47</font></span></a>, <a class="txt" href="#E21:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E97">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E27:81_1_1_1_1_2_1"/>
 <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="c6">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#K17" title="SUPINF_2:func.17">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="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E92">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></span>;<br/>
<a NAME="E28:81_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A53</font></span>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c14">n2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c14">n2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E98">A51</font></span></a>, <a class="txt" href="#E26:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E99">A52</font></span></a>, <a class="txt" href="#E17:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E95">A48</font></span></a>, <a class="ref" href="xxreal_3.html#T64" title="XXREAL_3:th.64">XXREAL_3:64</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:81_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E101">A54</font></span>: 
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span><br/><a NAME="E2:81_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A55</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a></span>;<br/><a NAME="E3:81_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E1:81_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E101">A54</font></span></a></span>;<br/><a NAME="E4:81_1_1_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c14">n2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E102">A55</font></span></a>, <a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/><a NAME="E5:81_1_1_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c14">n2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E81">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:81_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:81_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E100">A53</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:81_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="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:81_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A56</font></span>: 
<font color="Maroon" title="c3">L</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D9" target="_self" title="MESFUNC5:def.9">Def9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:81_1_1_1_1_2"/>
<font color="Maroon" title="c3">L</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span><br/><a NAME="E5:81_1_1_1_1_2"/><span class="kw">then </span>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1_1_1_2"><span class="lab"><font color="Green" title="E87">A56</font></span></a>, <a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:81_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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1"><span class="lab"><font color="Green" title="E85">A9</font></span></a>, <a class="txt" href="#E1:81_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A39</font></span></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:81_1_1_2"/><span class="lab"><font color="Green" title="E85">A57</font></span>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">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="81_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>3</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:81_1_1_2_1"/><span class="lab"><font color="Green" title="E86">A58</font></span>: 
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:81_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#K17" title="SUPINF_2:func.17">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#K17" title="SUPINF_2:func.17">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="txt" href="mesfunc5.html#E80"><span class="lab"><font color="Green" title="E75">Lm8</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E87">A59</font></span>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">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#K17" title="SUPINF_2:func.17">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> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">p92</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="c7">p9</font> = <font color="Maroon" title="c4">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E4:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E88">A60</font></span>: 
<font color="Maroon" title="c5">supk</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">p92</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#K17" title="SUPINF_2:func.17">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></span>;<br/><a NAME="E5:81_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A61</font></span>: 
<font color="Maroon" title="c7">p9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">supk</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">p92</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#K17" title="SUPINF_2:func.17">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></span>;<br/><a NAME="E6:81_1_1_2_1_1_1"/><span class="kw">then </span>
<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#K17" title="SUPINF_2:func.17">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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_2"><span class="lab"><font color="Green" title="E85">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">j</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><span class="kw"> such that </span><br/><a NAME="E8:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E90">A62</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font>
 <span class="kw">and </span><br/><a NAME="E9:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E91">A63</font></span>: 
<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#K17" title="SUPINF_2:func.17">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="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T94" title="XXREAL_2:th.94">XXREAL_2:94</a></span>;<br/><a NAME="E10:81_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">supk</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">p92</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="c5">supk</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">p92</font></span>)</span>
 ;<br/><a NAME="E11:81_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A64</font></span>: 
<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#K17" title="SUPINF_2:func.17">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#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">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="c7">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A60</font></span></a>, <a class="txt" href="#E5:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A61</font></span></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">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> <span class="kw"> such that </span><br/><a NAME="E13:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E93">A65</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E14:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E94">A66</font></span>: 
<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#K17" title="SUPINF_2:func.17">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="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_2_1"><span class="lab"><font color="Green" title="E86">A58</font></span></a>, <a class="txt" href="#E1:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A59</font></span></a>, <a class="ref" href="measure6.html#T6" title="MEASURE6:th.6">MEASURE6:6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">n1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E95">A67</font></span>: 
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E17:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E96">A68</font></span>: 
<font color="Maroon" title="c9">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="c10">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E93">A65</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">n2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E19:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E97">A69</font></span>: 
<font color="Maroon" title="c11">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">J</font>
 <span class="kw">and </span><br/><a NAME="E20:81_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E98">A70</font></span>: 
<font color="Maroon" title="c8">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="c11">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E90">A62</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">n1</font> = <font color="Maroon" title="c10">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E95">A67</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">n2</font> = <font color="Maroon" title="c11">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E97">A69</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">n</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>);<br/><a NAME="E23:81_1_1_2_1_1_1"/>
<font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E24:81_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A71</font></span>: 
<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#K17" title="SUPINF_2:func.17">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E91">A63</font></span></a>, <a class="txt" href="#E20:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A70</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E25:81_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E26:81_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A72</font></span>: 
<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#K17" title="SUPINF_2:func.17">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E94">A66</font></span></a>, <a class="txt" href="#E17:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E96">A68</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E27:81_1_1_2_1_1_1"/>
 <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#K17" title="SUPINF_2:func.17">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></span>;<br/><a NAME="E28:81_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A73</font></span>: 
 <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A72</font></span></a>, <a class="txt" href="#E24:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A71</font></span></a>, <a class="txt" href="#E11:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E92">A64</font></span></a>, <a class="ref" href="xxreal_3.html#T64" title="XXREAL_3:th.64">XXREAL_3:64</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:81_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E102">A74</font></span>: 
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">m</font></span><br/><a NAME="E2:81_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A75</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a></span>;<br/><a NAME="E3:81_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E1:81_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E102">A74</font></span></a></span>;<br/><a NAME="E4:81_1_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E103">A75</font></span></a>, <a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/><a NAME="E5:81_1_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> (<font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E81">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:81_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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:81_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A73</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E30:81_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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:81_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#K17" title="SUPINF_2:func.17">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:81_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E87">A76</font></span>: 
 <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="txt" href="#E1:81_1_1_2_1"><span class="lab"><font color="Green" title="E86">A58</font></span></a>, <a class="ref" href="mesfunc5.html#T59" target="_self" title="MESFUNC5:th.59">Th65</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">n2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:81_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E88">A77</font></span>: 
 <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="txt" href="#E1:81_1_1_2"><span class="lab"><font color="Green" title="E85">A57</font></span></a>, <a class="txt" href="#E1:81_1_1_2_1"><span class="lab"><font color="Green" title="E86">A58</font></span></a>, <a class="ref" href="mesfunc5.html#T59" target="_self" title="MESFUNC5:th.59">Th65</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">set </span><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="E7:81_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E89">A78</font></span>: 
<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="E8:81_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E9:81_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A79</font></span>: 
 <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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E87">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:81_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E11:81_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A80</font></span>: 
 <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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A77</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:81_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></span>;<br/><a NAME="E13:81_1_1_2_1_1_2"/><span class="kw">then </span>
<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#K3" title="SUPINF_2:func.3">+</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="nat_1.html#K6" title="NAT_1:func.6">.</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#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E91">A80</font></span></a>, <a class="txt" href="#E9:81_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E90">A79</font></span></a>, <a class="ref" href="xxreal_3.html#T64" title="XXREAL_3:th.64">XXREAL_3:64</a></span>;<br/><a NAME="E14:81_1_1_2_1_1_2"/><span class="kw">then </span>
<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="nat_1.html#K6" title="NAT_1:func.6">.</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#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A78</font></span></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></span>;<br/><a NAME="E15:81_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A81</font></span>: 
 <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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E81">A5</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:81_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E93">A82</font></span>: 
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span><br/><a NAME="E2:81_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A83</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a></span>;<br/><a NAME="E3:81_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E1:81_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E93">A82</font></span></a></span>;<br/><a NAME="E4:81_1_1_2_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E94">A83</font></span></a>, <a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/><a NAME="E5:81_1_1_2_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E81">A5</font></span></a></span>;<br/><a NAME="E6:81_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E81">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:81_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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:81_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E92">A81</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E17:81_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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E3:81_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A84</font></span>: 
<font color="Maroon" title="c3">L</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D9" target="_self" title="MESFUNC5:def.9">Def9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:81_1_1_2"/>
<font color="Maroon" title="c3">L</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span><br/><a NAME="E5:81_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A85</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1_2"><span class="lab"><font color="Green" title="E86">A84</font></span></a>, <a class="ref" href="mesfunc5.html#D12" target="_self" title="MESFUNC5:def.12">Def12</a></span>;<br/><a NAME="E6:81_1_1_2"/><span class="lab"><font color="Green" title="E88">A86</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E82">A6</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a>, <a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></span>;<br/><a NAME="E7:81_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> <font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="ref" href="mesfunc5.html#D5" target="_self" title="MESFUNC5:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:81_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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_2"><span class="lab"><font color="Green" title="E85">A57</font></span></a>, <a class="txt" href="#E5:81_1_1_2"><span class="lab"><font color="Green" title="E87">A85</font></span></a>, <a class="txt" href="#E6:81_1_1_2"><span class="lab"><font color="Green" title="E88">A86</font></span></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:81"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="SUPINF_2:func.17">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#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E85">A87</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:81_2"/><span class="lab"><font color="Green" title="E86">A88</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font></span><br/><a NAME="E2:81_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A89</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a></span>;<br/><a NAME="E3:81_2"/>
<font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E1:81_2"><span class="lab"><font color="Green" title="E86">A88</font></span></a></span>;<br/><a NAME="E4:81_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_2"><span class="lab"><font color="Green" title="E87">A89</font></span></a>, <a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/><a NAME="E5:81_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E81">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:81_2"/>
<font color="Maroon" title="c3">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E81">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:81"/>
 <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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#T54" target="_self" title="MESFUNC5:th.54">Th60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E12:81"/>
 <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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="mesfunc5.html#T54" target="_self" title="MESFUNC5:th.54">Th60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:81"/>
(  <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#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">J</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E8:81"><span class="lab"><font color="Green" title="E84">A8</font></span></a>, <a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E85">A87</font></span></a>, <a class="ref" href="mesfunc5.html#T54" target="_self" title="MESFUNC5:th.54">Th60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
