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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">T1</font>, <font color="Olive" title="b3">T2</font>, <font color="Olive" title="b4">T3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Olive" title="b1">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b5">F</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b5">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> holds <br/><font color="Olive" title="b5">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Olive" title="b2">T1</font>,<font color="Olive" title="b3">T2</font></span>)</span>,<font color="Olive" title="b4">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Olive" title="b2">T1</font>,<span class="p1">(<span class="default"><font color="Olive" title="b5">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Olive" title="b3">T2</font>,<font color="Olive" title="b4">T3</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T1</font>, <font color="Olive" title="b2">T2</font>, <font color="Olive" title="b3">T3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b4">F</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b4">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> holds <br/><font color="Olive" title="b4">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Olive" title="b1">T1</font>,<font color="Olive" title="b2">T2</font></span>)</span>,<font color="Olive" title="b3">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Olive" title="b1">T1</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Olive" title="b2">T2</font>,<font color="Olive" title="b3">T3</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c4">T2</font>, <font color="Maroon" title="c5">T3</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> holds <br/><font color="Olive" title="b1">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>,<font color="Maroon" title="c5">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c5">T3</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>,<font color="Maroon" title="c5">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c5">T3</font></span>)</span> )</span><br/>











<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c6">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>,<font color="Maroon" title="c5">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c5">T3</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1"/>
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><b>suppose </b></a><a NAME="E1:38_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>,<font color="Maroon" title="c5">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c5">T3</font></span>)</span></span><br/><div class="add"><a NAME="E2:38_1_1"/><b>then </b>
( <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c5">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="txt" href="finseqop.html#E12"><i><font color="Green" title="E7">Lm2</font></i></a></i>;<br/><a NAME="E3:38_1_1"/><b>then </b>
( <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>,<font color="Maroon" title="c5">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c5">T3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T87" title="FINSEQ_2:th.87">FINSEQ_2:87</a></i>;<br/><b>hence </b><a NAME="E4:38_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>,<font color="Maroon" title="c5">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c5">T3</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2"><b>suppose </b></a><a NAME="E1:38_1_2"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>,<font color="Maroon" title="c5">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c5">T3</font></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">C</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">i</font> as  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>  <b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><a NAME="E3:38_1_2"/>
( <font color="Maroon" title="c3">T1</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c4">T2</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c5">T3</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="txt" href="finseqop.html#E29"><i><font color="Green" title="E14">Lm5</font></i></a></i>;<br/><b>hence </b><a NAME="E4:38_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>,<font color="Maroon" title="c5">T3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c3">T1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c5">T3</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseqop.html#E1:38"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="funcop_1.html#T76" title="FUNCOP_1:th.76">FUNCOP_1:76</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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