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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="E1:1"/><i><font color="Green">E2</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <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">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> ;<br/>
<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> 0 = 
<font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="txt" href="measure7.html#E1:1"><i><font color="Green">E2</font></i></a></i>
;<br/>
<a NAME="E3:1"/><b>then </b><i><font color="Green">E3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_2"/>
<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>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/>

<b>hence </b><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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="txt" href="measure7.html#E1:1"><i><font color="Green">E2</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="measure7.html#E3:1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="measure7.html#E4:1"><i><font color="Green">E4</font></i></a>);<br/></i>
<a NAME="E6:1"/><i><font color="Green">E6</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T85">SUPINF_1:85</a></i>;<br/>
<a NAME="E7:1"/>
 <a href="supinf_2.html#K17">rng</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="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_3"><b>proof </b></a><div class="add">

<a NAME="E1:1_3"/><i><font color="Green">E7</font></i>: 
(  <a href="relset_1.html#K4">dom</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="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="supinf_2.html#K17">rng</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="tarski.html#R1">c=</a>  <a href="numbers.html#K6">ExtREAL</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:1_3"/>
 <a href="supinf_2.html#K17">rng</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="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="domain_1.html#K6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</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>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_3_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</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="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="measure7.html#E3:1_3_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E5:1_3_1"/>
<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>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E5:1"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:1_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="measure7.html#E3:1_3_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</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> 0
 
<b>by </b><i><a class="txt" href="measure7.html#E5:1"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</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>
 <b>by </b><i><a class="txt" href="measure7.html#E1:1_3"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:1"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure7.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="supinf_2.html#D23">SUPINF_2:def 23</a></i>;<br/>


</div>
