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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>thus </b><a NAME="E1:15"/>
( <a href="ami_2.html#K5" target="_self">SCM-OK</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a>  implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E7</font></i>: 
<a href="ami_2.html#K5" target="_self">SCM-OK</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a> 
 ;<br/>

<b>assume </b><a NAME="E2:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:15_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 
<b>by </b><i><a class="txt" href="ami_2.html#E10"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:15_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ami_2.html#E1:15_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="ami_2.html#D5" target="_self">Def5</a>, <a class="ref" href="ami_2.html#T6" target="_self">Th6</a></i>;<br/>


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

<b>thus </b><a NAME="E2:15"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <a href="ami_2.html#K5" target="_self">SCM-OK</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="ref" href="ami_2.html#D5" target="_self">Def5</a></i>;<br/>


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