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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b3">g</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="b3">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b3">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">G</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) holds <br/><font color="Olive" title="b3">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Olive" title="b1">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Olive" title="b2">G</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</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">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b1">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) holds <br/><font color="Olive" title="b1">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c3">G</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c4">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) implies <font color="Maroon" title="c4">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c3">G</font></span>)</span> )</span><br/>







<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>] <span class="kw">means </span>for <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">g</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="b2">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b2">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) holds <br/><font color="Olive" title="b2">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Olive" title="b1">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> $1</span>)</span>;<br/>
<a NAME="E1:15"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
for <font color="Olive" title="b1">G</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">G</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c6">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:15_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</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">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b1">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) holds <br/><font color="Olive" title="b1">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">g</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c8">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) implies <font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:15_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c8">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 <span class="kw">and </span><br/><a NAME="E3:15_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
( <font color="Maroon" title="c8">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E14">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:15_1_1"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><span class="kw">assume </span><a NAME="E2:15_1_1"/>
not <font color="Maroon" title="c8">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1</span><br/><span class="kw">hence </span><a NAME="E3:15_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E15">A7</font></span>: <font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
<br/>.= 
<font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="finsop_1.html#T5" target="_self" title="FINSOP_1:th.5">Th5</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_1_3_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="15_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_3_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:15_1_3_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E5:15_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c5">G</font></span>)</span>,<font color="Maroon" title="c6">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_3_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="finsop_1.html#T5" target="_self" title="FINSOP_1:th.5">Th5</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:15_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">hence </span><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c6">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="finsop_1.html#T5" target="_self" title="FINSOP_1:th.5">Th5</a></span>
<br/>.= 
<font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="finsop_1.html#E9"><span class="lab"><font color="Green" title="E5">Lm4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_3_1_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:15_1"/>
<font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:15"/><span class="lab"><font color="Green" title="E11">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</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">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b1">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) holds <br/><font color="Olive" title="b1">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c5">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c6">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) implies <font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c5">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_2"/>
<font color="Maroon" title="c6">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 <span class="kw">and </span><br/><a NAME="E2:15_2"/><span class="lab"><font color="Green" title="E12">A11</font></span>: 
( <font color="Maroon" title="c6">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c5">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c5">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c6">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c5">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c6">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_2"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></span>
<br/>.= 
<font color="Maroon" title="c6">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c5">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_2"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="txt" href="finsop_1.html#E8"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:15"/>
for <font color="Olive" title="b1">G</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S2" title="FINSEQ_2:sch.2">FINSEQ_2:sch 2</a>(<a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E10">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:15"/>
( <font color="Maroon" title="c4">g</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c4">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) implies <font color="Maroon" title="c4">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c3">G</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
