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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>,   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>) <b>-&gt; </b>   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> = <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">ADD</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E2:1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds  <font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E3:1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">ADD</font> is <a href="binop_1.html#V1">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">b</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>thus </b><font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">a</font> <a href="euclid.html#K7">+</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:1_1"/>
<font color="Maroon">ADD</font> is <a href="binop_1.html#V1">commutative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:1"/>
<font color="Maroon">ADD</font> is <a href="binop_1.html#V2">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">b</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">c</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font>, <font color="Maroon">c1</font> = <font color="Maroon">c</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>thus </b><font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">a</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K7">+</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">c1</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T32">RVSUM_1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:1_2"/>
<font color="Maroon">ADD</font> is <a href="binop_1.html#V2">associative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:1"/>
 ex <font color="Olive">ADD</font> being  <a href="binop_1.html#NM2">BinOp</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> st <br/>( ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds  <font color="Olive">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="euclid.html#K7">+</a> <font color="Olive">b</font> ) &amp; <font color="Olive">ADD</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Olive">ADD</font> is <a href="binop_1.html#V2">associative</a> )
 <b>by </b><i>, </i>;<br/>


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