<?xml version="1.0"?>
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<b>let </b><font color="Maroon">R1</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">R2</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>



<b>set </b><font color="Maroon">G</font> =  <a href="vectsp_2.html#K9" target="_self">BiModule</a> <font color="Maroon">R1</font>,<font color="Maroon">R2</font>;<br/>
<b>set </b><font color="Maroon">a</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R1</font>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R1</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R2</font>, <font color="Maroon">q</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R2</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp_2.html#K9" target="_self">BiModule</a> <font color="Maroon">R1</font>,<font color="Maroon">R2</font></span>)</span>, <font color="Maroon">w</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp_2.html#K9" target="_self">BiModule</a> <font color="Maroon">R1</font>,<font color="Maroon">R2</font></span>)</span>;<br/>





<a NAME="E1:72"/>
( <font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:72"/>
( <font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 ;<br/>

<a NAME="E3:72"/>
( <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">q</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:72"/>
( <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">q</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font> &amp; <font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">p</font> )
 ;<br/>


</div>
