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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></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="2_1_1"><b>suppose </b></a><a NAME="E1:2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2"><b>suppose </b></a><a NAME="E1:2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:2_1_2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E4:2_1_2"/><i><font color="Green">E5</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E3:2_1_2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/><a NAME="E5:2_1_2"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="measure7.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a>, <a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/><a NAME="E6:2_1_2"/><b>then </b>
( not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T12">XXREAL_0:12</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><a NAME="E7:2_1_2"/><b>then </b>
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure7.html#E4:2_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><b>hence </b><a NAME="E8:2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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