<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be  non <a href="struct_0.html#V3">empty</a> <a href="scmring1.html#V1" target="_self">good</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/>



<b>thus </b><a NAME="E1:19"/>
( <span class="p1">(<span class="default"><a href="scmring1.html#K2" target="_self">SCM-OK</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">G</font> implies  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="scmring1.html#K2" target="_self">SCM-OK</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">G</font>
 ;<br/>

<b>assume </b><a NAME="E2:19_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds not <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or  ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmring1.html#K2" target="_self">SCM-OK</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:19_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_1_2"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:19_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E4:19_1_1_2"/>
<span class="p1">(<span class="default"><a href="scmring1.html#K2" target="_self">SCM-OK</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:19_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:19"/>
(  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 implies <span class="p1">(<span class="default"><a href="scmring1.html#K2" target="_self">SCM-OK</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>


</div>
