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<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="fsm_2.html#V1" target="_self">calculating_type</a> <a href="fsm_2.html#V5" target="_self">halting</a>  <a href="fsm_2.html#L2" target="_self">Moore-SM_Final</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>;<br/>

<b>assume </b><a NAME="E1:50"/><i><font color="Green">E38</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <a href="numbers.html#K1">REAL</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> &amp; the <a href="fsm_2.html#U1" target="_self">FinalS</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; the <a href="fsm_1.html#U4">OFun</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> holds <br/>the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> holds <br/>the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<a NAME="E2:50"/>
(  <a href="square_1.html#K1">min</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="fsm_2.html#R2" target="_self">leads_to_final_state_of</a> <font color="Maroon">M</font> &amp;  <a href="square_1.html#K1">min</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="fsm_2.html#R3" target="_self">is_result_of</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">M</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:50"/>
 <a href="fsm_2.html#K2" target="_self">Result</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


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