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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font>, <font color="Maroon" title="c3">x3</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> ;<br/><b>thus </b><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">x1</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span>)</span> = 
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c3">x3</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c3">x3</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x3</font></span>)</span>

<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x3</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">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#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x3</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></i>
<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">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#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">x3</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></i>
;<br/><b>thus </b><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span>)</span>,<font color="Maroon" title="c3">x3</font> = 
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<font color="Maroon" title="c3">x3</font>
<b>by </b><i><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x3</font>
<b>by </b><i><a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x3</font></span>)</span>

<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x3</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">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#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">x3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x3</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></i>
<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">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#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">x3</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13"/>
 <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a>  <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> 
 <b>by </b><i><a class="ref" href="binop_1.html#T23" title="BINOP_1:th.23">BINOP_1:23</a></i>;<br/>


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