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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</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="Olive" title="b2">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a>  )</span><br/>





<a NAME="E1:34"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> $1</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> ;<br/>
<span class="kw">assume </span><a NAME="E2:34"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> </span><br/>

<div><span class="lab"><font color="Green" title="E32">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:34_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><a NAME="E2:34_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="polynom8.html#T1" target="_self" onmouseover="rs('polynom8/T1')" onmouseout="rh()">Th1</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">n</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="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D7" onmouseover="rs('group_1/D7')" onmouseout="rh()">GROUP_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="polynom8.html#T2" target="_self" onmouseover="rs('polynom8/T2')" onmouseout="rh()">Th2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 1</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">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T8" onmouseover="rs('binom/T8')" onmouseout="rh()">BINOM:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T10" onmouseover="rs('binom/T10')" onmouseout="rh()">BINOM:10</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:34_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T8" onmouseover="rs('binom/T8')" onmouseout="rh()">BINOM:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T8" onmouseover="rs('binom/T8')" onmouseout="rh()">BINOM:8</a></span>
;<br/>
<a NAME="E5:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E6:34"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E32">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:34"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
