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<b>let </b><font color="Maroon">J</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>, <font color="Maroon">K</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>, <font color="Maroon">L</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <b>and </b><br/><a NAME="E3:88"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">J</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp; <font color="Maroon">K</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> )
 <b>and </b><br/><a NAME="E4:88"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E5:88"/><i><font color="Green">E40</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">J</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:88_1_1"/>
(  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1"><b>suppose </b></a><a NAME="E1:88_1_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">SJ</font> =  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:88_1_1_1_1"/>
(  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:88_1_1_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">SK</font> =  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E3:88_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span>
 <b>by </b><i/>;<br/><div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:88_1_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:88_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/><b>consider </b><font color="Maroon">SJ'</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E4:88_1_1_1_1_1_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">SJ'</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font> &amp; <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">SJ'</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, , <a class="ref" href="measure6.html#T33">MEASURE6:33</a></i>;<br/><b>consider </b><font color="Maroon">nj</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:88_1_1_1_1_1_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">nj</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">J</font> &amp; <font color="Maroon">SJ'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">nj</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:88_1_1_1_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">nj</font> = <font color="Maroon">nj</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E4:88_1_1_1_1_1_1"><i><font color="Green">E95</font></i></a></i>;<br/><b>consider </b><font color="Maroon">SK'</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E9:88_1_1_1_1_1_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">SK'</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font> &amp; <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">SK'</font> )
 <b>by </b><i>, , <a class="ref" href="measure6.html#T33">MEASURE6:33</a></i>;<br/><b>consider </b><font color="Maroon">nk</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:88_1_1_1_1_1_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">nk</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font> &amp; <font color="Maroon">SK'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">nk</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">nk</font> = <font color="Maroon">nk</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E13:88_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E14:88_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">n</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">nj</font>,<font color="Maroon">nk</font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:88_1_1_1_1_1_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:88_1_1_1_1_1_1_1"/>
(  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E4:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
(  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span> )
 <b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E5:88_1_1_1_1_1_1_1"/>
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E6:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font> )
 <b>by </b><i><a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/><a NAME="E7:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="txt" href="mesfunc5.html#E11:88_1_1_1_1_1_1"><i><font color="Green">E100</font></i></a>, , <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/><a NAME="E8:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_2.html#T8">SUPINF_2:8</a>, <a class="ref" href="supinf_2.html#T9">SUPINF_2:9</a></i>;<br/><a NAME="E9:88_1_1_1_1_1_1_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:88_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span> &amp; <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="txt" href="mesfunc5.html#E4:88_1_1_1_1_1_1_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="mesfunc5.html#E14:88_1_1_1_1_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="mesfunc5.html#E11:88_1_1_1_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E11:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:88_1_1_1_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="extreal1.html#T40">EXTREAL1:40</a></i>;<br/><a NAME="E12:88_1_1_1_1_1_1_1"/><i><font color="Green">E109</font></i>: 
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T15">EXTREAL2:15</a></i>;<br/><a NAME="E13:88_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:88_1_1_1_1_1_1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="extreal1.html#T25">EXTREAL1:25</a></i>;<br/><a NAME="E14:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal2.html#T55">EXTREAL2:55</a></i>;<br/><i><font color="Green">E122</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">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">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:88_1_1_1_1_1_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="extreal1.html#T11">EXTREAL1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="extreal1.html#T11">EXTREAL1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">SK</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SJ</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="txt" href="mesfunc5.html#E14:88_1_1_1_1_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="mesfunc5.html#E11:88_1_1_1_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="extreal2.html#T40">EXTREAL2:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">SK</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SJ</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E16:88_1_1_1_1_1_1_1"/>
(  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">SJ'</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">SK'</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:88_1_1_1_1_1_1"><i><font color="Green">E94</font></i></a>, , <a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/><a NAME="E17:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
( <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SJ'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="extreal2.html#T30">EXTREAL2:30</a></i>;<br/><a NAME="E18:88_1_1_1_1_1_1_1"/>
<font color="Maroon">nk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E19:88_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">nk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:88_1_1_1_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E20:88_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">nk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E21:88_1_1_1_1_1_1_1"/><b>then </b>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">SK'</font> &amp; <font color="Maroon">SK'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, , <a class="txt" href="mesfunc5.html#E1:88_1_1_1_1_1_1_1"><i><font color="Green">E102</font></i></a>, , <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">s0</font> = <font color="Maroon">SK'</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><b>reconsider </b><font color="Maroon">s1</font> =  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E24:88_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E25:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SJ'</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:88_1_1_1_1_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="measure6.html#T12">MEASURE6:12</a></i>;<br/><a NAME="E26:88_1_1_1_1_1_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E27:88_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:88_1_1_1_1_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="measure6.html#T12">MEASURE6:12</a></i>;<br/><a NAME="E28:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SJ'</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:88_1_1_1_1_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/><a NAME="E29:88_1_1_1_1_1_1_1"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">SJ'</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">SK'</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="txt" href="mesfunc5.html#E4:88_1_1_1_1_1_1"><i><font color="Green">E95</font></i></a>, , </i>;<br/><a NAME="E30:88_1_1_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SJ'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_2.html#T10">SUPINF_2:10</a></i>;<br/><a NAME="E31:88_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E126</font></i>: 
( <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SJ'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/><a NAME="E32:88_1_1_1_1_1_1_1"/>
( <font color="Maroon">nj</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">nk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E33:88_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">nj</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">nk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:88_1_1_1_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E34:88_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">SJ'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">SK'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="txt" href="mesfunc5.html#E4:88_1_1_1_1_1_1"><i><font color="Green">E95</font></i></a>, </i>;<br/><a NAME="E35:88_1_1_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SJ'</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/><a NAME="E36:88_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SJ</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SJ'</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">SK</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">SK'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><b>hence </b><a NAME="E37:88_1_1_1_1_1_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon">SJ</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">SK</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:88_1_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="mesfunc5.html#E6:88_1_1_1_1_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="mesfunc5.html#E8:88_1_1_1_1_1_1_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div><a NAME="E5:88_1_1_1_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E6:88_1_1_1_1_1"/>
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E7:88_1_1_1_1_1"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, , , <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:88_1_1_1_1_2"/><i><font color="Green">E128</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:88_1_1_1_1_2"/>
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:88_1_1_1_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:88_1_1_1_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E4:88_1_1_1_1_2_1"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font> &amp; <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="measure6.html#T33">MEASURE6:33</a></i>;<br/>
<b>consider </b><font color="Maroon">n1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:88_1_1_1_1_2_1"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">J</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">supj</font> =  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p'2</font> = <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2, <font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E10:88_1_1_1_1_2_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">supj</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p'2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/>
<a NAME="E11:88_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">p'</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">supj</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p'2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/>
<a NAME="E12:88_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E14:88_1_1_1_1_2_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="hahnban.html#T8">HAHNBAN:8</a></i>;<br/>
<b>consider </b><font color="Maroon">n2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:88_1_1_1_1_2_1"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n2</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">n</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font>;<br/>
<a NAME="E18:88_1_1_1_1_2_1"/>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> )
 
<b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/>
<a NAME="E19:88_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E20:88_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E135</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, , , , <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>
<a NAME="E21:88_1_1_1_1_2_1"/><i><font color="Green">E136</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<a NAME="E22:88_1_1_1_1_2_1"/><i><font color="Green">E137</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>
<a NAME="E23:88_1_1_1_1_2_1"/>
<font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E24:88_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E138</font></i>: 
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<a NAME="E25:88_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E139</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> &amp; <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, , <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>
<a NAME="E26:88_1_1_1_1_2_1"/>
<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">supj</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p'2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">supj</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p'2</font></span>)</span>
 
;<br/>
<a NAME="E27:88_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, , <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/>
<a NAME="E28:88_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E140</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:88_1_1_1_1_2_1_1"/>
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:88_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:88_1_1_1_1_2_1_1"/>
<font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:88_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/><a NAME="E6:88_1_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E7:88_1_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:88_1_1_1_1_2_1_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E30:88_1_1_1_1_2_1"/>
 ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:88_1_1_1_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E4:88_1_1_1_1_2"/>
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:88_1_1_1_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E6:88_1_1_1_1_2"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><b>hence </b><a NAME="E7:88_1_1_1_1_2"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_2"><b>suppose </b></a><a NAME="E1:88_1_1_2"/><i><font color="Green">E184</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:88_1_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:88_1_1_2_1"/><b>then </b><i><font color="Green">E185</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:88_1_1_2_1_1"/>
(  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:88_1_1_2_1_1_1"/><i><font color="Green">E186</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:88_1_1_2_1_1_1"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font> &amp; <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="measure6.html#T33">MEASURE6:33</a></i>;<br/><b>consider </b><font color="Maroon">n1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:88_1_1_2_1_1_1"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">supk</font> =  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a></i>;<br/><b>reconsider </b><font color="Maroon">p'2</font> = <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2, <font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E9:88_1_1_2_1_1_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">supk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p'2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E10:88_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">p'</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">supk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p'2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E11:88_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E13:88_1_1_2_1_1_1"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="hahnban.html#T8">HAHNBAN:8</a></i>;<br/><b>consider </b><font color="Maroon">n2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:88_1_1_2_1_1_1"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">J</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n2</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>set </b><font color="Maroon">n</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font>;<br/><a NAME="E17:88_1_1_2_1_1_1"/>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> )
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E18:88_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E19:88_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E193</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/><a NAME="E20:88_1_1_2_1_1_1"/><i><font color="Green">E194</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E21:88_1_1_2_1_1_1"/><i><font color="Green">E195</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/><a NAME="E22:88_1_1_2_1_1_1"/>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E23:88_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E196</font></i>: 
( <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E24:88_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E197</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, , <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/><a NAME="E25:88_1_1_2_1_1_1"/>
<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">supk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p'2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">supk</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p'2</font></span>)</span>
 ;<br/><a NAME="E26:88_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E27:88_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E198</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:88_1_1_2_1_1_1_1"/>
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:88_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E199</font></i>: 
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:88_1_1_2_1_1_1_1"/>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:88_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/><a NAME="E6:88_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E7:88_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:88_1_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E29:88_1_1_2_1_1_1"/>
 ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:88_1_1_2_1_1_2"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:88_1_1_2_1_1_2"/><i><font color="Green">E200</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a></i>;<br/><b>consider </b><font color="Maroon">n2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:88_1_1_2_1_1_2"/><i><font color="Green">E201</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a></i>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1</font>, <font color="Maroon">n2</font> = <font color="Maroon">n2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon">n</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font>;<br/><a NAME="E7:88_1_1_2_1_1_2"/>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> )
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E8:88_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E9:88_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E202</font></i>: 
(  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, , <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/><a NAME="E10:88_1_1_2_1_1_2"/>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E11:88_1_1_2_1_1_2"/><b>then </b>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E12:88_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E203</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:88_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E14:88_1_1_2_1_1_2"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E15:88_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E204</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:88_1_1_2_1_1_2_1"/>
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:88_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E205</font></i>: 
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E3:88_1_1_2_1_1_2_1"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E4:88_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E5:88_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E6:88_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:88_1_1_2_1_1_2_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:88_1_1_2_1_1_2"/>
 ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:88_1_1_2"/><b>then </b><i><font color="Green">E206</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E4:88_1_1_2"/>
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:88_1_1_2"/><b>then </b><i><font color="Green">E207</font></i>: 
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E6:88_1_1_2"/>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:88_1_1_2"/><b>then </b>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> 0 &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, , <a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/><b>hence </b><a NAME="E8:88_1_1_2"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:88"/>
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 ;<br/>

<div><i><font color="Green">E208</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:88_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:88_2"/><b>then </b><i><font color="Green">E209</font></i>: 
( <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E3:88_2"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E4:88_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E5:88_2"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:88_2"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:88"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>

<a NAME="E10:88"/>
(  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">J</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>
<b>hence </b><a NAME="E11:88"/>
(  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">J</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K</font></span>)</span> &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">J</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>


</div>
