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<a NAME="E1:12_1_1"/><i><font color="Green">E10</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></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><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V3">constant</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>thus </b><a NAME="E4:12_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="lopban_3.html#E1:12_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="lopban_3.html#E1:12_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E3:12_1_1"/><i><font color="Green">E11</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>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:12_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_1_1_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>hence </b><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lopban_3.html#E1:12_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="lopban_3.html#E3:12_1_1"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="lopban_3.html#E1:12_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="lopban_3.html#E3:12_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="lopban_3.html#E1:12_1_1_2"><i><font color="Green">E12</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:12_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>;<br/>


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