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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  =  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> $1</span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> $2</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:44"/><i><font color="Green" title="E27">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fsm_2.html#L2" title="FSM_2:struct.2">Moore-SM_Final</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><a NAME="E3:44"/><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c2">M</font> is <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; the <a href="fsm_1.html#U4" title="FSM_1:sel.4">OFun</a> of <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> )
 ;<br/>

<b>assume </b><a NAME="E4:44"/><i><font color="Green" title="E29">A3</font></i>: 
( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st ( <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 ) holds <br/>the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 holds <br/>the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 holds <br/>the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">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" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><a NAME="E1:44_1"/>
the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">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" title="c5">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E30">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E31">A5</font></i>: 
 <a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="c6">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="c6">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:44"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="fsm_2.html#E2:44_1_1_1_1_1"><i><font color="Green" title="E31">A5</font></i></a></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" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="fsm_2.html#E2:44_1_1_1_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:44"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_1"><i><font color="Green" title="E30">A4</font></i></a></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" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="fsm_2.html#E2:44_1_1_1_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:44"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_1"><i><font color="Green" title="E30">A4</font></i></a></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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">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" title="E30">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E31">A7</font></i>: 
 <a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="c6">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E32">A8</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="fsm_2.html#E2:44_1_1_1_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:44"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_2_1_1_1"><i><font color="Green" title="E32">A8</font></i></a></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" title="E32">A9</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:44"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_2_1_1_2"><i><font color="Green" title="E32">A9</font></i></a></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" title="E32">A10</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="fsm_2.html#E2:44_1_1_1_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:44"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_2_1_1_3"><i><font color="Green" title="E32">A10</font></i></a></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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">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" title="E30">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E31">A12</font></i>: 
 <a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="c6">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E32">A13</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="fsm_2.html#E2:44_1_1_1_1_3"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:44"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_3_1_1_1"><i><font color="Green" title="E32">A13</font></i></a></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" title="E32">A14</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="fsm_2.html#E2:44_1_1_1_1_3"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:44"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_3"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_3_1_1_2"><i><font color="Green" title="E32">A14</font></i></a></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" title="E32">A15</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D2" title="ABSVALUE:def.2">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:44"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_3"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="fsm_2.html#E2:44_1_1_1_1_3"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="fsm_2.html#E1:44_1_1_1_1_3_1_1_3"><i><font color="Green" title="E32">A15</font></i></a></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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c6">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" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c2">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="fsm_2.html#E2:44"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:44"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="fsm_2.html#R3" title="FSM_2:pred.3">is_result_of</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E3:44"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="fsm_2.html#T22" target="_self" title="FSM_2:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E7:44"/>
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="absvalue.html#K2" title="ABSVALUE:func.2">sgn</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="fsm_2.html#R3" title="FSM_2:pred.3">is_result_of</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="fsm_2.html#E2:44"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>


</div>
