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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:37"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> &amp; ( ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 ) or <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>suppose </b></a><a NAME="E1:37_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>set </b><font color="Maroon">e</font> =  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>;<br/><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_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="E1:37_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>;<br/><a NAME="E3:37_1_1_1_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="finsop_1.html#T11">FINSOP_1:11</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>;<br/><b>hence </b><a NAME="E3:37_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="finsop_1.html#T11">FINSOP_1:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:37_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1"/>
<font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finsop_1.html#T11">FINSOP_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2"><b>suppose </b></a><a NAME="E1:37_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies <font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> );<br/><a NAME="E2:37_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><a NAME="E3:37_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:37_1_2_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies <font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_2_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_2_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a></i>;<br/><b>hence </b><a NAME="E3:37_1_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_2_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:37_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E4:37_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span> = 
<font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span>

<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default">1 <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T17">FINSOP_1:17</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>hence </b><a NAME="E6:37_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E4:37_1_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><b>hence </b><a NAME="E5:37_1_2"/>
<font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="finsop_1.html#K1">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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