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<b>let </b><font color="Maroon">D</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">e</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <font color="Maroon">F</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>;<br/>
<a NAME="E3:28"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E4:28"/><i><font color="Green">E56</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span> &amp;  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">p</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">q</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> = 
<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">q</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_1_2"/><i><font color="Green">E58</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">p</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">q</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">q</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</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="E2:28_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funcop_1.html#K6">.:</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">p</font>,<font color="Maroon">e</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">q</font>,<font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T48">FUNCOP_1:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28"/>
<font color="Maroon">F</font> <a href="funcop_1.html#K6">.:</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">p</font>,<font color="Maroon">e</font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">q</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="setwop_2.html#K1" target="_self">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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