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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">d</font>, <font color="Maroon" title="c3">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c5">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/>









<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:30_1"/><i><font color="Green" title="E16">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></i>;<br/><a NAME="E2:30_1"/><i><font color="Green" title="E17">A3</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c5">p</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> or  not <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">i</font></span>)</span> = 
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="ref" href="setwop_2.html#T22" target="_self" title="SETWOP_2:th.22">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:30_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="setwop_2.html#E2:30_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="setwop_2.html#E1:30_1_1_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="funcop_1.html#T42" title="FUNCOP_1:th.42">FUNCOP_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c5">p</font></span>)</span>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:30_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="setwop_2.html#E2:30_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="setwop_2.html#E1:30_1_1_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="setwop_2.html#T22" target="_self" title="SETWOP_2:th.22">Th22</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_2"/><i><font color="Green" title="E18">A5</font></i>: 
not <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">i</font></span>)</span> = 
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c3">e</font>
<b>by </b><i><a class="ref" href="setwop_2.html#T22" target="_self" title="SETWOP_2:th.22">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c5">p</font></span>)</span>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:30"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="setwop_2.html#E1:30_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="setwop_2.html#E2:30_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="setwop_2.html#E1:30_1_1_1_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="setwop_2.html#T22" target="_self" title="SETWOP_2:th.22">Th22</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:30_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> <font color="Maroon" title="c2">d</font>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">e</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c5">p</font></span>)</span>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T66" title="FUNCOP_1:th.66">FUNCOP_1:66</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
<font color="Maroon" title="c4">F</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> <font color="Maroon" title="c2">d</font>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c5">p</font></span>)</span>,<font color="Maroon" title="c3">e</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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