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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<font color="Maroon">a<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E2:114"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> = the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<font color="Maroon">a<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:114"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<font color="Maroon">a<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E6:114"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> = the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<font color="Maroon">a<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:114"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>16</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = 0;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = 1;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T128">ZFMISC_1:128</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font>;<br/>
<b>take </b><font color="Maroon">c<sub>26</sub></font> =  <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>27</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T128">ZFMISC_1:128</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>30</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>27</sub></font> ;<br/>
<b>take </b><font color="Maroon">c<sub>32</sub></font> =  <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>31</sub></font> #);<br/>

<a NAME="E17:114"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E17:114"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>
;<br/>
<b>hence </b><a NAME="E19:114"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>26</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>thus </b><a NAME="E20:114"/>
<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:114_2"/><i><font color="Green">E89</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>26</sub></font> &amp; ( for <font color="Olive">b<sub>1</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>26</sub></font> holds <br/><font color="Maroon">c<sub>33</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">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E3:114_2"/><i><font color="Green">E90</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:114_2_1"/><i><font color="Green">E91</font></i>: 
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:114_2_1_1"/>
(  not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> or  not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> )
 </span><b>by </b><i><a class="txt" href="fsm_1.html#E1:114_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_2_1_1_1"><b>suppose </b></a><a NAME="E1:114_2_1_1_1"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:114_2_1_1_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> &amp; not <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:114_2_1_1_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:114_2_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E5:114_2_1_1_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E7:114_2_1_1_1"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>hence </b><a NAME="E8:114_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E3:114_2_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E5:114_2_1_1_1"><i><font color="Green">E93</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_2_1_1_2"><b>suppose </b></a><a NAME="E1:114_2_1_1_2"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:114_2_1_1_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:114_2_1_1_2"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</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="E5:114_2_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:114_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:114_2_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:114_2_1_1_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E5:114_2_1_1_2"><i><font color="Green">E93</font></i></a></i>;<br/><b>hence </b><a NAME="E8:114_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:114_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E6:114_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>26</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>34</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="fsm_1.html#D19" target="_self">FSM_1:def 19</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>35</sub></font>, <font color="Maroon">c<sub>36</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:114_2_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> )
 ;<br/><a NAME="E2:114_2_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><a NAME="E3:114_2_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K2">`2</a>  )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E1:114_2_2"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E4:114_2_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E1:114_2_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E5:114_2_2"/>
<font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E1:114_2_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E6:114_2_2"/><b>then </b>
( <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:114_2_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:114_2_2"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:114_2_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="fsm_1.html#E2:114_2_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E3:114_2_2"><i><font color="Green">E93</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:114_2"/><b>then </b>
( <font color="Maroon">c<sub>34</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>34</sub></font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E3:114_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:114_2"/>
<font color="Maroon">c<sub>34</sub></font> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>

<b>thus </b><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>26</sub></font> = 
<font color="Maroon">c<sub>24</sub></font> <a href="domain_1.html#K3">`2</a> 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>35</sub></font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>26</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>36</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:114_2_4"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>38</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>36</sub></font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><a NAME="E4:114_2_4"/><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Maroon">c<sub>38</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:114_2_4"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>26</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>36</sub></font></span>)</span> = 
<font color="Maroon">c<sub>34</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">c<sub>36</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E2:114"><i><font color="Green">E84</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">c<sub>36</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E4:114_2_4"><i><font color="Green">E92</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">c<sub>36</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>,<font color="Maroon">c<sub>36</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a></i>
;<br/><b>thus </b>the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>26</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>36</sub></font> = 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">c<sub>36</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E4:114"><i><font color="Green">E85</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>,<font color="Maroon">c<sub>36</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E2:114_2"><i><font color="Green">E89</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:114_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>26</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>26</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>26</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> )
 ;<br/>


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

<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E22:114"/><i><font color="Green">E89</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>32</sub></font> &amp; ( for <font color="Olive">b<sub>1</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>32</sub></font> holds <br/><font color="Maroon">c<sub>33</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">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E23:114"/><i><font color="Green">E90</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:114_3"/><i><font color="Green">E91</font></i>: 
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:114_3_1"/>
(  not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> or  not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> )
 </span><b>by </b><i><a class="txt" href="fsm_1.html#E1:114_3"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_3_1_1"><b>suppose </b></a><a NAME="E1:114_3_1_1"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:114_3_1_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> &amp; not <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:114_3_1_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:114_3_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E5:114_3_1_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E7:114_3_1_1"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>hence </b><a NAME="E8:114_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E3:114_3_1_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E5:114_3_1_1"><i><font color="Green">E93</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_3_1_2"><b>suppose </b></a><a NAME="E1:114_3_1_2"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:114_3_1_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:114_3_1_2"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</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="E5:114_3_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:114_3_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:114_3_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:114_3_1_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E5:114_3_1_2"><i><font color="Green">E93</font></i></a></i>;<br/><b>hence </b><a NAME="E8:114_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:114_3_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E6:114_3_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>32</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>34</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="fsm_1.html#D19" target="_self">FSM_1:def 19</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>35</sub></font>, <font color="Maroon">c<sub>36</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:114_4"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> )
 ;<br/><a NAME="E2:114_4"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><a NAME="E3:114_4"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K2">`2</a>  )
 <b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E1:114_4"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E4:114_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E1:114_4"><i><font color="Green">E91</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E5:114_4"/>
<font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E1:114_4"><i><font color="Green">E91</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E6:114_4"/><b>then </b>
( <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>36</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:114_4"><i><font color="Green">E94</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:114_4"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:114_4"><i><font color="Green">E91</font></i></a>, <a class="txt" href="fsm_1.html#E2:114_4"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E3:114_4"><i><font color="Green">E93</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E26:114"/><b>then </b>
( <font color="Maroon">c<sub>34</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>34</sub></font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E23:114"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E27:114"/>
<font color="Maroon">c<sub>34</sub></font> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>

<b>thus </b><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>32</sub></font> = 
<font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>35</sub></font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>32</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>36</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:114_6"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>set </b><font color="Maroon">c<sub>38</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>36</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/><a NAME="E3:114_6"/><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>36</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:114_6"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>32</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>36</sub></font></span>)</span> = 
<font color="Maroon">c<sub>34</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">c<sub>36</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E6:114"><i><font color="Green">E86</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">c<sub>36</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E3:114_6"><i><font color="Green">E92</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">c<sub>36</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>,<font color="Maroon">c<sub>36</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a></i>
;<br/><b>thus </b>the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>32</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>36</sub></font> = 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">c<sub>36</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E8:114"><i><font color="Green">E87</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>,<font color="Maroon">c<sub>36</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E22:114"><i><font color="Green">E89</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E30:114"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>32</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>32</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>32</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> )
 ;<br/>


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