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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>thus </b><a NAME="E1:18"/>
(  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</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="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</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="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="ring_1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:18"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:18"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="ring_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</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="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>


</div>
