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<span class="kw">thus </span><a NAME="E1:29_3_1"/>
 <a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a>  is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</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">  <a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a>  is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">r1</font>, <font color="Maroon" title="c2">r2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D2" onmouseover="rs('binop_1/D2')" onmouseout="rh()">BINOP_1:def 2</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"> <a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c1">r1</font>)</span><br/>

<span class="kw">thus </span><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>) = 
<font color="Maroon" title="c1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">r2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" target="_self" onmouseover="rs('binop_2/D9')" onmouseout="rh()">Def9</a></span>
<br/>.= 
<a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c1">r1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" target="_self" onmouseover="rs('binop_2/D9')" onmouseout="rh()">Def9</a></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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c1">r1</font>, <font color="Maroon" title="c2">r2</font>, <font color="Maroon" title="c3">r3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D3" onmouseover="rs('binop_1/D3')" onmouseout="rh()">BINOP_1:def 3</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"> <a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">r1</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r3</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>)</span>)</span>,<font color="Maroon" title="c3">r3</font>)</span><br/>

<span class="kw">thus </span><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c1">r1</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r3</font>)</span>)</span>) = 
<font color="Maroon" title="c1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r3</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" target="_self" onmouseover="rs('binop_2/D9')" onmouseout="rh()">Def9</a></span>
<br/>.= 
<font color="Maroon" title="c1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">r3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" target="_self" onmouseover="rs('binop_2/D9')" onmouseout="rh()">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">r3</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">r3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" target="_self" onmouseover="rs('binop_2/D9')" onmouseout="rh()">Def9</a></span>
<br/>.= 
<a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="binop_2.html#K9" title="BINOP_2:func.9">addreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>)</span>)</span>,<font color="Maroon" title="c3">r3</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" target="_self" onmouseover="rs('binop_2/D9')" onmouseout="rh()">Def9</a></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"> verum</span><br/>


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