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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1">Real</a>,  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:44"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</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="E3:44"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">M</font> is <a href="fsm_2.html#V1" target="_self">calculating_type</a> &amp; 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> )
 ;<br/>

<b>assume </b><a NAME="E4:44"/><i><font color="Green">E40</font></i>: 
( ( 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#NR4" target="_self">&gt;</a> 0 or <font color="Olive">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) 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> 1 ) &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="hidden.html#R1">=</a> 0 or <font color="Olive">y</font> <a href="hidden.html#R1">=</a> 0 ) &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 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> 0 ) &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> 0 &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;</a> 0 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>  <a href="real_1.html#K1">-</a> 1 ) )
 ;<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/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><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="E1:44_1"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1_1_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1_1_1_1_1_1_1"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_1_1_1_2"/><b>then </b>
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><a NAME="E3:44_1_1_1_1_1_1_1_2"/><b>then </b>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1_1_1_1_1_2"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_1_1_1_3"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_1_1_1_3"/><b>then </b>
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><a NAME="E3:44_1_1_1_1_1_1_1_3"/><b>then </b>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1_1_1_1_1_3"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:44_1_1_1_1_1"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_2_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_2_1_1_1"/><b>then </b>
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><a NAME="E3:44_1_1_1_1_2_1_1_1"/><b>then </b>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1_1_2_1_1_1"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_2_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_2_1_1_2"/><b>then </b>
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1_1_1_2_1_1_2"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_2_1_1_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_2_1_1_3"/><b>then </b>
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><a NAME="E3:44_1_1_1_1_2_1_1_3"/><b>then </b>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1_1_2_1_1_3"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:44_1_1_1_1_2"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_3"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1_1_1_3_1_1"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_3_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_3_1_1_1"/><b>then </b>
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><a NAME="E3:44_1_1_1_1_3_1_1_1"/><b>then </b>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1_1_3_1_1_1"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_3_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_3_1_1_2"/><b>then </b>
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><a NAME="E3:44_1_1_1_1_3_1_1_2"/><b>then </b>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1_1_3_1_1_2"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_3_1_1_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_3_1_1_3"/><b>then </b>
 <a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2">ABSVALUE:def 2</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1_1_1_3_1_1_3"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:44_1_1_1_1_3"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:44_1_1"/>
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="Maroon">x</font>,<font color="Maroon">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>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:44_1"/>
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="Maroon">x</font>,<font color="Maroon">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="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:44"/><b>then </b>
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</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>, </i>;<br/>
<b>hence </b><a NAME="E7:44"/>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2">sgn</a> <font color="Maroon">y</font></span>)</span> <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/>;<br/>


</div>
