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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span>;<br/>











<b>assume </b><a NAME="E1:28"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</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">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">a</font> &amp; <font color="Maroon">y</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">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">b</font> )
 ;<br/>

<b>consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:28"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</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">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">a1</font> &amp; <font color="Maroon">y</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">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">b1</font> )
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green">E48</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</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">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E5:28"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">b1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 
<b>by </b><i>, , </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T42">RLVECT_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T42">RLVECT_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
;<br/>
<a NAME="E7:28"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="ideal_1.html#D1">IDEAL_1:def 1</a></i>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="binop_1.html#D1">BINOP_1:def 1</a></i>
<br/>.= 
 <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">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>


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