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

<span class="kw">set </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">b</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</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">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> )</span><br/><span class="kw">assume </span><a NAME="E1:7_1_1_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c7">s</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E3:7_1_1_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c7">s</font>
 <span class="kw">and </span><br/><a NAME="E4:7_1_1_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c6">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:7_1_1_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="c7">s</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></span>;<br/><span class="lab"><font color="Green" title="E7">A5</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">s</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><div><span class="lab"><font color="Green" title="E8">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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">k</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="c7">s</font> implies <font color="Maroon" title="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> )</span><br/><span class="kw">assume </span><a NAME="E1:7_1_1_1_2"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c8">k</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="c7">s</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="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">b1</font>, <font color="Maroon" title="c10">b2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:7_1_1_1_2"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c10">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E4:7_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_1_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">b19</font>, <font color="Maroon" title="c12">b29</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E6:7_1_1_1_2"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c11">b19</font>,<font color="Maroon" title="c12">b29</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E7:7_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">b19</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c12">b29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_1_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E1:7_1_1_1_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="pre_poly.html#T68" title="PRE_POLY:th.68">PRE_POLY:68</a></span>;<br/><span class="lab"><font color="Green" title="E14">A12</font></span>: <font color="Maroon" title="c10">b2</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c11">b19</font>,<font color="Maroon" title="c12">b29</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1_1_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E6:7_1_1_1_2"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c12">b29</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>
;<br/><font color="Maroon" title="c9">b1</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c11">b19</font>,<font color="Maroon" title="c12">b29</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1_1_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E6:7_1_1_1_2"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c11">b19</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>
;<br/><a NAME="E10:7_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A13</font></span>: 
( <font color="Maroon" title="c9">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c10">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E7:7_1_1_1_2"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E8:7_1_1_1_2"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="pre_poly.html#T53" title="PRE_POLY:th.53">PRE_POLY:53</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_2_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="E1:7_1_1_1_2_3_1"/>
( <font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> or <font color="Maroon" title="c4">c2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:7_1_1_1_2"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="polynom7.html#D8" title="POLYNOM7:def.8">POLYNOM7:def 8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_2_3_1_1"><span class="kw">case </span></a><a NAME="E1:7_1_1_1_2_3_1_1"/>
<font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</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="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:7_1_1_1_2_3_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1_1_1_2"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="binom.html#T1" title="BINOM:th.1">BINOM:1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_2_3_1_2"><span class="kw">case </span></a><a NAME="E1:7_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c4">c2</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c10">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</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="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:7_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1_1_1_2"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="binom.html#T2" title="BINOM:th.2">BINOM:2</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:7_1_1_1_2"/>
<font color="Maroon" title="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ; <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="hs2(this)" href="javascript:()" title="7_1_1_1_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_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="E1:7_1_1_1_3_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_3_1_1"><span class="kw">case </span></a><a NAME="E1:7_1_1_1_3_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span><br/><div class="add"><a NAME="E2:7_1_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E3:7_1_1_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1_1_1"><span class="lab"><font color="Green" title="E5">A3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_3_1_2"><span class="kw">case </span></a><a NAME="E1:7_1_1_1_3_1_2"/><span class="lab"><font color="Green" title="E9">A14</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c8">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">s</font>;<br/><a NAME="E3:7_1_1_1_3_1_2"/><span class="lab"><font color="Green" title="E10">A15</font></span>: 
<font color="Maroon" title="c8">k</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="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_1_3_1_2"><span class="lab"><font color="Green" title="E9">A14</font></span></a></span>;<br/><a NAME="E4:7_1_1_1_3_1_2"/>
for <font color="Olive" title="b1">k9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k9</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="c7">s</font> &amp; <font color="Olive" title="b1">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">k</font> holds <br/><font color="Maroon" title="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_1_1_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a></span>;<br/><a NAME="E5:7_1_1_1_3_1_2"/><span class="kw">then </span>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1_1_3_1_2"><span class="lab"><font color="Green" title="E10">A15</font></span></a>, <a class="ref" href="polynom2.html#T5" title="POLYNOM2:th.5">POLYNOM2:5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:7_1_1_1_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E7:7_1_1_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E3:7_1_1_1_3_1_2"><span class="lab"><font color="Green" title="E10">A15</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:7_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ; <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="kw">hence </span><a NAME="E2:7_1_1"/>
<font color="Maroon" title="c3">c1</font> <a href="polynom1.html#K11" title="POLYNOM1:func.11">*'</a> <font color="Maroon" title="c4">c2</font> is  <a href="polynom7.html#V4" title="POLYNOM7:attr.4">Constant</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D8" title="POLYNOM7:def.8">POLYNOM7:def 8</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>
