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<b>let </b><font color="Maroon">G1</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> , <font color="Maroon">G2</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><b>that </b><br/><a NAME="E1:8_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">n</font> ) &amp; ( not <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 <b>and </b><br/><a NAME="E2:8_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">G2</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">n</font> ) &amp; ( not <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">G2</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 ;<br/>

<a NAME="E3:8_1_2"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">G1</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then 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>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_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="E1:8_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_2_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><a NAME="E2:8_1_2_1_1_1"/><b>then </b>
<font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:8_1_2_1_1_1"/>
<font color="Maroon">G1</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_2_1_1_2"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><a NAME="E2:8_1_2_1_1_2"/><b>then </b>
<font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:8_1_2_1_1_2"/>
<font color="Maroon">G1</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:8_1_2"/>
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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