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<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ami_1.html#NM3">Instruction</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:36"/>
 <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="ami_5.html#D1">AMI_5:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E3:36"/><i><font color="Green">E18</font></i>: 
( the <a href="ami_1.html#U1">Instruction-Counter</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; the <a href="ami_1.html#U3">Instruction-Codes</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp; the <a href="ami_1.html#U4">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="amistd_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>, <a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:36"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:36"/><i><font color="Green">E20</font></i>: 
the <a href="ami_1.html#U6">Execution</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p4">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="amistd_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E7:36"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="txt" href="amistd_1.html#E3:36"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E8:36"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="amistd_1.html#E2:36"><i><font color="Green">E17</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:36_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1"/>
contradiction
 <b>by </b><i><a class="txt" href="amistd_1.html#E2:36"><i><font color="Green">E17</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:36"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p4">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then </b><i><font color="Green">E23</font></i>: the <a href="ami_1.html#U6">Execution</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="amistd_1.html#E6:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="amistd_1.html#E9:36"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="amistd_1.html#E8:36"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E12:36"/><i><font color="Green">E24</font></i>: 
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="numbers.html#K5">NAT</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:36"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E14:36"/><i><font color="Green">E26</font></i>: 
 <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="amistd_1.html#E3:36"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="txt" href="amistd_1.html#E11:36"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="txt" href="amistd_1.html#E5:36"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="txt" href="amistd_1.html#E13:36"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E12:36"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E14:36"><i><font color="Green">E26</font></i></a></i>
;<br/><br/>


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