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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">d1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">d2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><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/><b>let </b><font color="Maroon">T</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <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>assume </b><a NAME="E1:41"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>suppose </b></a><a NAME="E1:41_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:41_1_1"/><b>then </b>
( <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <font color="Maroon">d2</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T113">FINSEQ_2:113</a></i>;<br/><a NAME="E3:41_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <font color="Maroon">d1</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <font color="Maroon">d2</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T93">FINSEQ_2:93</a></i>;<br/><b>hence </b><a NAME="E4:41_1_1"/>
<font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <font color="Maroon">d1</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <font color="Maroon">d2</font>,<font color="Maroon">T</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2"><b>suppose </b></a><a NAME="E1:41_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">C</font> =  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E3:41_1_2"/>
<font color="Maroon">T</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:41_1_2"/>
<font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <font color="Maroon">d1</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K3" target="_self">[;]</a> <font color="Maroon">d2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T77">FUNCOP_1:77</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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