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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">R</font><br/> for <font color="Olive" title="b2">p</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">R</font> holds   <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</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">R</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> 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">R</font> holds   <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Olive" title="b1">p</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">p</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">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span><br/>





<b>consider </b><font color="Maroon" title="c4">f</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">R</font><b> such that </b><br/><a NAME="E2:11"/><i><font color="Green" title="E5">A1</font></i>: 
(  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c1">R</font> &amp; ( for <font color="Olive" title="b1">j</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> <br/> for <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> ) )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">g</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">R</font><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green" title="E6">A2</font></i>: 
(  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c1">R</font> &amp; ( for <font color="Olive" title="b1">j</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> <br/> for <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c5">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> ) )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>;<br/>
<i><font color="Green" title="E7">A3</font></i>:  <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"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>
<br/>.= 
 <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> <font color="Maroon" title="c3">p</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green" title="E8">A4</font></i>:  <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"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></i>
<br/>.= 
 <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> <font color="Maroon" title="c3">p</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b><font color="Maroon" title="c4">f</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="c5">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1;<br/>
<a NAME="E7:11"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="#E2:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E4:11"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<div><i><font color="Green" title="E10">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">j</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">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="c6">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:11_3"/><i><font color="Green" title="E11">A7</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>thus </b><a NAME="E2:11_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_1"/><i><font color="Green" title="E12">A8</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">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="c6">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:11_3_1"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:11"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="#E1:11_3"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></i>;<br/>
<a NAME="E3:11_3_1"/><b>then </b><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c6">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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<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="E4:11_3_1"/><i><font color="Green" title="E15">A11</font></i>: 
<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="c6">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="E5:11_3_1"/><b>then </b>
<font color="Maroon" title="c6">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> <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"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:11_3_1"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:11_3_1"/><b>then </b>
<font color="Maroon" title="c6">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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:11_3_1"/><b>then </b><i><font color="Green" title="E16">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="E8:11_3_1"/><i><font color="Green" title="E17">A13</font></i>: 
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="#E1:11_3"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></i>;<br/>
<a NAME="E9:11_3_1"/>
<font color="Maroon" title="c6">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> <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"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="#E6:11"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="#E3:11_3_1"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="#E4:11_3_1"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:11_3_1"/><b>then </b><i><font color="Green" title="E18">A14</font></i>: 
<font color="Maroon" title="c6">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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:11_3_1"/><b>then </b><i><font color="Green" title="E19">A15</font></i>: 
<font color="Maroon" title="c6">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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>;<br/>
<a NAME="E12:11_3_1"/><i><font color="Green" title="E20">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="#E10:11_3_1"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c5">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="#E2:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E1:11_3_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="#E2:11_3_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="#E7:11_3_1"><i><font color="Green" title="E16">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><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="c6">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></span>)</span>
<b>by </b><i><a class="txt" href="#E11:11_3_1"><i><font color="Green" title="E19">A15</font></i></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="#E11:11_3_1"><i><font color="Green" title="E19">A15</font></i></a>, <a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c5">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="#E4:11"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="#E8:11_3_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="#E12:11_3_1"><i><font color="Green" title="E20">A16</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>
<a NAME="E9:11"/><i><font color="Green" title="E11">A17</font></i>: 
for <font color="Olive" title="b1">i</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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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">i</font>]
 
<b>from </b><i><a class="ref" href="polynom2.html#S4" title="POLYNOM2:sch.4">POLYNOM2:sch 4</a>(<a class="txt" href="#E7:11"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="#E8:11"><i><font color="Green" title="E10">A6</font></i></a>);<br/></i>
<b>thus </b> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> = 
<font color="Maroon" title="c4">f</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="#E2:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E6:11"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c5">g</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="#E9:11"><i><font color="Green" title="E11">A17</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="#E5:11"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</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>
