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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">ADD</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b1">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b3">b</font> ) &amp; <font color="Olive" title="b1">ADD</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Olive" title="b1">ADD</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )</span><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>) <b>-&gt; </b>   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> = $1 <a href="euclid.html#K7" title="EUCLID:func.7">+</a> $2;<br/>
<b>consider </b><font color="Maroon" title="c2">ADD</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E2:1"/><i><font color="Green" title="E2">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E3:1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c2">ADD</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">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" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><br/><b>thus </b><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> = 
<font color="Maroon" title="c3">a</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">b</font>
<b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E2">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E2">A1</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>
<b>hence </b><a NAME="E2:1_1"/>
<font color="Maroon" title="c2">ADD</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</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>
<a NAME="E4:1"/>
<font color="Maroon" title="c2">ADD</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">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" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">c</font></span><br/><b>reconsider </b><font color="Maroon" title="c6">a1</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">b1</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">c1</font> = <font color="Maroon" title="c5">c</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1" title="EUCLID:def.1">EUCLID:def 1</a></i>;<br/><b>thus </b><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E2">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E2">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c7">b1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c8">c1</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E2">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E2">A1</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>
<b>hence </b><a NAME="E2:1_2"/>
<font color="Maroon" title="c2">ADD</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</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>
<b>hence </b><a NAME="E5:1"/>
 ex <font color="Olive" title="b1">ADD</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b1">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b3">b</font> ) &amp; <font color="Olive" title="b1">ADD</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Olive" title="b1">ADD</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 <b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E3:1"><i><font color="Green" title="E3">A2</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>
