<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <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="Olive" title="b3">q</font> &amp;  ex <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b4">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="Olive" title="b2">p</font> &amp; <font color="Olive" title="b3">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> &amp; ( for <font color="Olive" title="b5">i'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">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="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b3">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i'</font> ) ) holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b2">p</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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">p</font>, <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <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="Olive" title="b2">q</font> &amp;  ex <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b3">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="Olive" title="b1">p</font> &amp; <font color="Olive" title="b2">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> &amp; ( for <font color="Olive" title="b4">i'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">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="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">i</font> holds <br/><font color="Olive" title="b2">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i'</font> ) ) holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">p</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <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="c4">q</font> &amp;  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">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="c3">p</font> &amp; <font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; ( for <font color="Olive" title="b2">i'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">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="c3">p</font> &amp; <font color="Olive" title="b2">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i'</font> ) ) implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">p</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E10">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <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="c4">q</font> &amp;  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">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="c3">p</font> &amp; <font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; ( for <font color="Olive" title="b2">i'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">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="c3">p</font> &amp; <font color="Olive" title="b2">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i'</font> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">p</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:16"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c5">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="c3">p</font> &amp; <font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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="c3">p</font> &amp; <font color="Olive" title="b1">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i'</font> ) )
 ;<br/>
<a NAME="E4:16"/><i><font color="Green" title="E12">A3</font></i>: 
(  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">L</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c3">p</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">L</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c4">q</font> )
 
<b>by </b><i><a class="ref" href="group_4.html#D3" title="GROUP_4:def.3">GROUP_4:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">p</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="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</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">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">p</font></span>)</span></span><br/><div class="add"><a NAME="E2:16_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>hence </b><a NAME="E3:16_1_1"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><a NAME="E1:16_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</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">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">p</font></span>)</span></span><br/><div class="add"><a NAME="E2:16_1_2"/><b>then </b><i><font color="Green" title="E13">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">fp</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:16_1_2"/><i><font color="Green" title="E14">A5</font></i>: 
( <font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> holds <br/><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">L</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:16"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="finsop_1.html#T2" title="FINSOP_1:th.2">FINSOP_1:2</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">fq</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:16_1_2"/><i><font color="Green" title="E15">A6</font></i>: 
( <font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">q</font> holds <br/><font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">L</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E4:16"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="#E2:16_1_2"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="finsop_1.html#T2" title="FINSOP_1:th.2">FINSOP_1:2</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( ( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 ) or ( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> ) );<br/><b>consider </b><font color="Maroon" title="c8">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:16_1_2"/><i><font color="Green" title="E16">A7</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">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> <font color="Maroon" title="c8">l</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E9:16_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Olive" title="b1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">l</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">i'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:16_1_2"/><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c9">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i'</font> &amp; <font color="Maroon" title="c9">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">l</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c10">l'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:16_1_2"/><i><font color="Green" title="E18">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">l'</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c8">l</font>, <font color="Maroon" title="c12">l'</font> = <font color="Maroon" title="c10">l'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><i><font color="Green" title="E19">A10</font></i>: <font color="Maroon" title="c11">l</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c12">l'</font>
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E13:16_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E16:16_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">l</font>
 <b>by </b><i><a class="txt" href="#E11:16_1_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:16_1_2"/><b>then </b><i><font color="Green" title="E20">A11</font></i>: 
( 1 <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="c3">p</font> &amp; 1 <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="c4">q</font> )
 <b>by </b><i><a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="#E13:16_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E15:16_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_2_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_2_1_1"><b>case </b></a><a NAME="E1:16_1_2_2_1_1"/><i><font color="Green" title="E21">A12</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</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">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span><br/><div class="add"><b>thus </b><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1 = 
<font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="#E4:16_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="#E17:16_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E17:16_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="#E1:16_1_2_2_1_1"><i><font color="Green" title="E21">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1
<b>by </b><i><a class="txt" href="#E6:16_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="#E17:16_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_2_1_2"><b>case </b></a><a NAME="E1:16_1_2_2_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:16_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E11:16_1_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1 = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E6:16_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="#E13:16_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E15:16_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E4:16_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="#E17:16_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/> <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></div><b>end;</b></div><a NAME="E19:16_1_2"/><b>then </b><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 ;<br/><a NAME="E20:16_1_2"/><i><font color="Green" title="E22">A14</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:16_1_2_3"/><i><font color="Green" title="E23">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E2:16_1_2_3"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_3_1"/>
( <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_3_1_1"/><i><font color="Green" title="E25">A17</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_3_1_1_1_1"/>
( <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_3_1_1_1_1_1"><b>case </b></a><a NAME="E1:16_1_2_3_1_1_1_1_1"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:16_1_2_3_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E3:16_1_2_3_1_1_1_1_1"/>
<font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="#E13:16_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E15:16_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="#E1:16_1_2_3_1_1_1_1_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>then </b><font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E6:16_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="#E1:16_1_2_3"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="#E2:16_1_2_3"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="#E1:16_1_2_3_1_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="txt" href="#E1:16_1_2_3_1_1_1_1_1"><i><font color="Green" title="E26">A18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="#E4:16_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="#E1:16_1_2_3"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="#E2:16_1_2_3_1_1_1_1_1"><i><font color="Green" title="E27">A19</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:16_1_2_3_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="#E1:16_1_2_3_1_1_1_1_1"><i><font color="Green" title="E26">A18</font></i></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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_3_1_1_1_1_2"><b>case </b></a><a NAME="E1:16_1_2_3_1_1_1_1_2"/><i><font color="Green" title="E26">A20</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:16_1_2_3_1_1_1_1_2"/>
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="#E1:16_1_2_3_1_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:16_1_2_3_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A21</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">l</font>
 <b>by </b><i><a class="txt" href="#E11:16_1_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:16_1_2_3_1_1_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:16_1_2_3_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E28">A22</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">l</font>
 <b>by </b><i><a class="txt" href="#E3:16_1_2_3_1_1_1_1_2"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:16_1_2_3_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E29">A23</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E7:16_1_2_3_1_1_1_1_2"/>
<font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E13:16_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E15:16_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="#E5:16_1_2_3_1_1_1_1_2"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>then </b><i><font color="Green" title="E30">A24</font></i>: <font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E6:16_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="#E1:16_1_2_3"><i><font color="Green" title="E23">A15</font></i></a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">L</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="#E1:16_1_2_3_1_1_1_1_2"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="#E5:16_1_2_3_1_1_1_1_2"><i><font color="Green" title="E28">A22</font></i></a>, <a class="txt" href="#E6:16_1_2_3_1_1_1_1_2"><i><font color="Green" title="E29">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E4:16_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="#E1:16_1_2_3"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="#E2:16_1_2_3"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="#E1:16_1_2_3_1_1"><i><font color="Green" title="E25">A17</font></i></a></i>
;<br/><a NAME="E9:16_1_2_3_1_1_1_1_2"/>
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_3_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_2_3_1_1_1_1_2_2"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_2_3_1_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:16_1_2_3_1_1_1_1_2"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_2_3_1_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_2_3_1_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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><b>hence </b><a NAME="E10:16_1_2_3_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="#E8:16_1_2_3_1_1_1_1_2"><i><font color="Green" title="E30">A24</font></i></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></div><b>end;</b></div><b>hence </b><a NAME="E3:16_1_2_3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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="16_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_3_1_2"/><i><font color="Green" title="E25">A25</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:16_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E26">A26</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:16_1_2_3_1_2"/>
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l</font>
 <b>by </b><i><a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="#E1:16_1_2_3"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:16_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E27">A27</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">l</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:16_1_2_3_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:16_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E28">A28</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c3">p</font>
 <b>by </b><i><a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="#E4:16_1_2_3_1_2"><i><font color="Green" title="E27">A27</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:16_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E29">A29</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E8:16_1_2_3_1_2"/>
<font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="#E13:16_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E15:16_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="#E6:16_1_2_3_1_2"><i><font color="Green" title="E28">A28</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>then </b><font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">fq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E6:16_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="#E1:16_1_2_3"><i><font color="Green" title="E23">A15</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E2:16_1_2_3"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="#E1:16_1_2_3_1_2"><i><font color="Green" title="E25">A25</font></i></a>, <a class="txt" href="#E2:16_1_2_3_1_2"><i><font color="Green" title="E26">A26</font></i></a>, <a class="txt" href="#E6:16_1_2_3_1_2"><i><font color="Green" title="E28">A28</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fp</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:16_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="#E1:16_1_2_3"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="#E7:16_1_2_3_1_2"><i><font color="Green" title="E29">A29</font></i></a></i>
;<br/><b>hence </b><a NAME="E10:16_1_2_3_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="#E1:16_1_2_3_1_2"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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>

</div><b>end;</b></div><a NAME="E21:16_1_2"/><i><font color="Green" title="E23">A30</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S4" target="_self" title="POLYNOM2:sch.4">POLYNOM2:sch 4</a>(<a class="txt" href="#E19:16_1_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E20:16_1_2"><i><font color="Green" title="E22">A14</font></i></a>);<br/></i><a NAME="E22:16_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l</font>
 <b>by </b><i><a class="txt" href="#E8:16_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E23:16_1_2"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:16_1_2"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="#E4:16_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="#E6:16_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="#E11:16_1_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="#E21:16_1_2"><i><font color="Green" title="E23">A30</font></i></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>

</div>
