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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>thus </b><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
;<br/><b>thus </b><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> = 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:5"/>
 <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#R6">is_distributive_wrt</a>  <a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i><a class="ref" href="binop_1.html#T23">BINOP_1:23</a></i>;<br/>


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