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<div class="add">

<b>let </b><font color="Maroon">L</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:86"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:86"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:86_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:86_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:86_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:86_1"/>
<font color="Maroon">n</font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:86_1"/>
<font color="Maroon">v</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:86"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<a NAME="E5:86"/>
( 1 <a href="hidden.html#R2">in</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/>
<a NAME="E6:86"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_2"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c</font> <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#T17" target="_self">Th17</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1"><b>suppose </b></a><a NAME="E1:86_2_1"/>
<font color="Maroon">c</font> <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">rc</font> = <font color="Maroon">c</font> 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><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_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="E1:86_2_1_1"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_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="E1:86_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:86_2_1_1_1"/>
<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> <font color="Maroon">rc</font></span>)</span> <a href="hidden.html#R1">=</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> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><font color="Maroon">L</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#T16" target="_self">Th16</a></i>;<br/><a NAME="E3:86_2_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> <font color="Maroon">rc</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>;<br/><b>hence </b><a NAME="E4:86_2_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> <font color="Maroon">rc</font></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="ref" href="extreal2.html#T53">EXTREAL2:53</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:86_2_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/><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="Olive">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> <font color="Maroon">rc</font></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>
 ;<br/></div><b>end;</b></div><a NAME="E4:86_2_1"/><b>then </b><i><font color="Green">E44</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="E5:86_2_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="E6:86_2_1"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, , <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E7:86_2_1"/>
 <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#T19" target="_self">Th19</a>, <a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="supinf_1.html#T80">SUPINF_1:80</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_2"><b>suppose </b></a><a NAME="E1:86_2_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:86_2_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 <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> 0 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="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_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="E1:86_2_2_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/>

<a NAME="E2:86_2_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">p</font> <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/>
<b>take </b>
0
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_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="E1:86_2_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:86_2_2_1_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, <a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E3:86_2_2_1_1"/>
<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> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:86_2_2_1"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:86_2_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:86_2_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/><b>hence </b><a NAME="E5:86_2_2"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, , <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E6:86_2_2"/>
 <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#T19" target="_self">Th19</a>, <a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="supinf_1.html#T80">SUPINF_1:80</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_3"><b>suppose </b></a><a NAME="E1:86_2_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:86_2_3"/>
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="86_2_3_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="E1:86_2_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:86_2_3_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">p</font> <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/>
<b>take </b>
0
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_3_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="E1:86_2_3_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:86_2_3_1_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</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#T16" target="_self">Th16</a>, <a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E3:86_2_3_1_1"/>
<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> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:86_2_3_1"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st 0 <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="E3:86_2_3"/><b>then </b><i><font color="Green">E100</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="E4:86_2_3"/>
<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="E5:86_2_3"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, , <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E6:86_2_3"/>
 <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#T19" target="_self">Th19</a>, <a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a>, <a class="ref" href="supinf_1.html#T80">SUPINF_1:80</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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