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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bcialg_4.html#NM1" title="BCIALG_4:NM.1">BCI-Algebra_with_Condition(S)</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="bcialg_4.html#U1" title="BCIALG_4:sel.1">ExternalDiff</a> of <font color="Maroon" title="c1">X</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="bcialg_4.html#U1" title="BCIALG_4:sel.1">ExternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bcialg_4.html#U1" title="BCIALG_4:sel.1">ExternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>)</span><br/><span class="kw">thus </span> the <a href="bcialg_4.html#U1" title="BCIALG_4:sel.1">ExternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) = 
<font color="Maroon" title="c2">a</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">b</font>

<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></span>
<br/>.= 
 the <a href="bcialg_4.html#U1" title="BCIALG_4:sel.1">ExternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>)

; <a class="txt" 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">hence </span><a NAME="E2:30"/>
 the <a href="bcialg_4.html#U1" title="BCIALG_4:sel.1">ExternalDiff</a> of <font color="Maroon" title="c1">X</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></span>; <a class="txt" 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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