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<b>let </b><font color="Maroon">I</font> be   <a href="scmpds_4.html#NM1">Program-block</a>, <font color="Maroon">J</font> be   <a href="scmpds_4.html#NM1">Program-block</a>;<br/>

<a NAME="E1:44"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:44"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<i><font color="Green">E31</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:44_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a>  )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1"><b>suppose </b></a><a NAME="E1:44_2_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:44_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> 
 <b>by </b><i/>;<br/><a NAME="E3:44_2_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:44_2_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ami_3.html#T34">AMI_3:34</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="scmpds_4.html#E2:44"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_2"><b>suppose </b></a><a NAME="E1:44_2_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> 
 ;<br/><div class="add"><a NAME="E2:44_2_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2">inspos</a> 0
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:44"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K2" target="_self">Initialized</a> <font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_4.html#K2" target="_self">Initialized</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K3" target="_self">';'</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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