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<span class="kw">thus </span><a NAME="E1:34_8_1"/>
 <a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_8_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">i2</font>,<font color="Maroon" title="c1">i1</font></span><br/>

<span class="kw">thus </span><a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> = 
<font color="Maroon" title="c1">i1</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">i2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D22" target="_self" title="BINOP_2:def.22">Def22</a></span>
<br/>.= 
<a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">i2</font>,<font color="Maroon" title="c1">i1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D22" target="_self" title="BINOP_2:def.22">Def22</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">i1</font>, <font color="Maroon" title="c2">i2</font>, <font color="Maroon" title="c3">i3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">i2</font>,<font color="Maroon" title="c3">i3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font></span>)</span>,<font color="Maroon" title="c3">i3</font></span><br/>

<span class="kw">thus </span><a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">i2</font>,<font color="Maroon" title="c3">i3</font></span>)</span> = 
<font color="Maroon" title="c1">i1</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">i2</font>,<font color="Maroon" title="c3">i3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D22" target="_self" title="BINOP_2:def.22">Def22</a></span>
<br/>.= 
<font color="Maroon" title="c1">i1</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i2</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c3">i3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D22" target="_self" title="BINOP_2:def.22">Def22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">i2</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c3">i3</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c3">i3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D22" target="_self" title="BINOP_2:def.22">Def22</a></span>
<br/>.= 
<a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K46" title="BINOP_2:func.46">multint</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font></span>)</span>,<font color="Maroon" title="c3">i3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D22" target="_self" title="BINOP_2:def.22">Def22</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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