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<b>let </b><font color="Maroon">L</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>, <font color="Maroon">K</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:89"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:89"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 <b>and </b><br/><a NAME="E3:89"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>

<a NAME="E4:89"/><i><font color="Green">E39</font></i>: 
(  <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">c</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</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#T31">SUPINF_1:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_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">L</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">L</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="89_1_1_1"><b>suppose </b></a><a NAME="E1:89_1_1_1"/><i><font color="Green">E40</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">L</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><a NAME="E2:89_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <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">L</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="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#T31">SUPINF_1:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:89_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font>
 ;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:89_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">m</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">x</font> <a href="hidden.html#R1">=</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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:89_1_1_1_1"/><i><font color="Green">E43</font></i>: 
(  <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/><b>reconsider </b><font color="Maroon">m</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/>;<br/><a NAME="E6:89_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</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>, </i>;<br/><a NAME="E7:89_1_1_1_1"/><i><font color="Green">E47</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">L</font></span>)</span> 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#D16">SUPINF_1:def 16</a></i>;<br/><a NAME="E8:89_1_1_1_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <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#T24" target="_self">Th24</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E9:89_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="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="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/><b>hence </b><a NAME="E10:89_1_1_1_1"/>
<font color="Maroon">x</font> <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">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T42">EXTREAL1:42</a></i>;<br/></div><b>end;</b></div><a NAME="E4:89_1_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></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#D9">SUPINF_1:def 9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:89_1_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">y</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:89_1_1_1_2"/><i><font color="Green">E98</font></i>: 
(  <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>  &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>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_1_1_2_1"/>
(  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 or  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:89_1_1_1_2_1_1"/><i><font color="Green">E99</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:89_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/><a NAME="E3:89_1_1_1_2_1_1"/>
<font color="Maroon">K</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">K</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E4:89_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> 1</span>)</span> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <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></i>;<br/><b>hence </b><a NAME="E5:89_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:89_1_1_1"><i><font color="Green">E94</font></i></a>, , <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:89_1_1_1_2_1_2"/><i><font color="Green">E101</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:89_1_1_1_2_1_2_1"/>
<font color="Maroon">x</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="E3:89_1_1_1_2_1_2_1"/><i><font color="Green">E102</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> &amp; <font color="Maroon">x</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/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</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#E4:89_1_1_1"><i><font color="Green">E94</font></i></a></i>;<br/><a NAME="E5:89_1_1_1_2_1_2_1"/>
( <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E6:89_1_1_1_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <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></i>;<br/><a NAME="E7:89_1_1_1_2_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">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T48">EXTREAL1:48</a></i>;<br/><b>hence </b><a NAME="E8:89_1_1_1_2_1_2_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:89_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a>, , <a class="ref" href="extreal2.html#T19">EXTREAL2:19</a></i>;<br/></div><b>end;</b></div><a NAME="E3:89_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> 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="E4:89_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E103</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">L</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/><div><i><font color="Green">E104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_1_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_1_1_2_1_2_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><a NAME="E2:89_1_1_1_2_1_2_2"/>
<font color="Maroon">K</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">K</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:89_1_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-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></i>;<br/><a NAME="E4:89_1_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/><a NAME="E5:89_1_1_1_2_1_2_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i/>;<br/><a NAME="E6:89_1_1_1_2_1_2_2"/>
<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="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:89_1_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> 1 <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">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/><a NAME="E8:89_1_1_1_2_1_2_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">L</font> <a href="mesfunc1.html#K4">.</a> 1 &amp; <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> 1 <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="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/><b>hence </b><a NAME="E9:89_1_1_1_2_1_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="extreal1.html#T28">EXTREAL1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_2_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_1_1_2_1_2_3"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 </span><b>by </b><i><a class="txt" href="mesfunc5.html#E1:89_1_1_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_2_1_2_3_1"><b>suppose </b></a><a NAME="E1:89_1_1_1_2_1_2_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_1_1_2_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_2_1_2_3_2"><b>suppose </b></a><a NAME="E1:89_1_1_1_2_1_2_3_2"/>
<font color="Maroon">y</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">ry</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>reconsider </b><font color="Maroon">sl</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">L</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/><a NAME="E4:89_1_1_1_2_1_2_3_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ry</font> <a href="real_1.html#K6">/</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T32">EXTREAL1:32</a></i>;<br/><a NAME="E5:89_1_1_1_2_1_2_3_2"/><b>then </b>
<font color="Maroon">sl</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ry</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, , <a class="txt" href="mesfunc5.html#E4:89_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xreal_1.html#T85">XREAL_1:85</a></i>;<br/><b>hence </b><a NAME="E6:89_1_1_1_2_1_2_3_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:89_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#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_2"><b>suppose </b></a><a NAME="E1:89_1_1_2"/><i><font color="Green">E108</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">L</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="hsNdiv(this)" href="javascript:()" title="89_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_1_2_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:89_1_1_2_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:89_1_1_2_1_1_1"/>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:89_1_1_2_1_1_1"/>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div><a NAME="E3:89_1_1_2_1_1"/><b>then </b>
(  <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> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 0 )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a></i>;<br/><b>hence </b><a NAME="E4:89_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#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:89_1_1_2_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:89_1_1_2_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, , <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:89_1_1_2_1_2_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</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="mesfunc5.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E2:89_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<a href="supinf_1.html#K2">-infty</a>  <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</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> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T44">EXTREAL1:44</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_2_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="E1:89_1_1_2_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font> or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:89_1_1_2_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:89_1_1_2_1_2_1_1_1"/>
<a href="supinf_1.html#K2">-infty</a>  <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><a NAME="E4:89_1_1_2_1_2_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">K</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></i>;<br/><b>hence </b><a NAME="E5:89_1_1_2_1_2_1_1_1"/>
 <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>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:89_1_1_2_1_2_1_1_2"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><a NAME="E2:89_1_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:89_1_1_2_1_2_1_1_2"/>
 <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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:89_1_1_2_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">K</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</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="E1:89_1_1_2_1_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:89_1_1_2_1_2_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:89_1_1_2_1_2_2"/><i><font color="Green">E125</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E5:89_1_1_2_1_2_2"/><i><font color="Green">E126</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">k</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</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> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T44">EXTREAL1:44</a></i>;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</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/><span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="real_1.html#K6">/</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>
<br/>.= 
<font color="Maroon">k1</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K6">/</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T83">XCMPLX_1:83</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="xcmplx_1.html#T51">XCMPLX_1:51</a></i>
;<br/><a NAME="E8:89_1_1_2_1_2_2"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">k</font> <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>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:89_1_1_2_1_2_2"/>
not for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <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>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:89_1_1_2_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> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</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#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:89"/>
 <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> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.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">L</font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:89_2"/><i><font color="Green">E127</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</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="mesfunc5.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E2:89_2"/>
( <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  implies <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></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="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a></i>;<br/><a NAME="E3:89_2"/><b>then </b><i><font color="Green">E128</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"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T28">EXTREAL1:28</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font> or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_2_1_1"><b>suppose </b></a><a NAME="E1:89_2_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:89_2_1_1"/>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E4:89_2_1_1"/>
 <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>
 <b>by </b><i><a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_2_1_2"><b>suppose </b></a><a NAME="E1:89_2_1_2"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><a NAME="E2:89_2_1_2"/><b>then </b>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:89_2_1_2"/>
 <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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:89"/>
<font color="Maroon">K</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 <b>by </b><i/>;<br/>


</div>
