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<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="arytm_2.html#K2">REAL+</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>1</sub></font> <a href="arytm_2.html#K8">*'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_1.html#T26">ARYTM_1:26</a></i>
<br/>.= 
 <a href="arytm_3.html#K13">{}</a> 
<b>by </b><i><a class="txt" href="arytm_0.html#E2:8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="arytm_1.html#T18">ARYTM_1:18</a></i>
;<br/><b>then </b> <a href="arytm_3.html#K13">{}</a>  = 
<font color="Maroon">c<sub>2</sub></font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:8"><i><font color="Green">E11</font></i></a>, <a class="ref" href="arytm_1.html#T2">ARYTM_1:2</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:8_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E4:8_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>1</sub></font> <a href="arytm_2.html#K8">*'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="arytm_1.html#K1">-'</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="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_1.html#T26">ARYTM_1:26</a></i>
<br/>.= 
 <a href="arytm_3.html#K13">{}</a> 
<b>by </b><i><a class="txt" href="arytm_0.html#E2:8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="arytm_1.html#T18">ARYTM_1:18</a></i>
;<br/><b>then </b> <a href="arytm_3.html#K13">{}</a>  = 
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:8"><i><font color="Green">E11</font></i></a>, <a class="ref" href="arytm_1.html#T2">ARYTM_1:2</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:8_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E4:8_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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