<?xml version="1.0"?>
<div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
<br/>.= 
<font color="Maroon">a'</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b'</font>
<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#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font>

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

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/><i><font color="Green">E19</font></i>: <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
;<br/><i><font color="Green">E22</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> = 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
;<br/><i><font color="Green">E23</font></i>: <font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
;<br/><i><font color="Green">E24</font></i>: <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> = 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><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#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font></span>)</span>,<font color="Maroon">c'</font> = 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b'</font></span>)</span>,<font color="Maroon">c'</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">a'</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b'</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">c'</font>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<font color="Maroon">a'</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c'</font></span>)</span>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a'</font>,<span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c'</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> <font color="Maroon">a'</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">b'</font>,<font color="Maroon">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
;<br/><b>hence </b><a NAME="E7:11_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/><a NAME="E1:11_1_1_3"/><i><font color="Green">E39</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><i><font color="Green">E40</font></i>: <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> </span>)</span> = 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> </span>)</span>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,the <a href="struct_0.html#U2">Zero</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,0
<b>by </b><i>, <a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>
;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">t</font> = 0 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a'</font>,<font color="Maroon">t</font> = 
<font color="Maroon">a'</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<font color="Maroon">a</font>

;<br/><b>hence </b><a NAME="E6:11_1_1_3"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> = 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">v</font>

<br/>.= 
<font color="Maroon">a'</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">v'</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> 
<b>by </b><i/>
;<br/></div>
<b>end;</b></div>

<a NAME="E5:11_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  holds  <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> = 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a'</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:11_1_1_5"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:11_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  holds  <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/>

<i><font color="Green">E46</font></i>: <font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
;<br/>
<a NAME="E2:11_1_1_6"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:11_1_1_6"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:11_1_1_6"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E50</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>

<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font>, <font color="Maroon">c</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">t'</font> = <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a</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">b</font>,<font color="Maroon">c</font></span>)</span> = 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c</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">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c</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">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c</font></span>)</span>

<br/>.= 
<font color="Maroon">t</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">t'</font>
<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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E10:11_1_1_6"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/><b>thus </b><a NAME="E1:11_1_1_7"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E8:11_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  holds  <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16">GROUP_1:def 16</a><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/><a NAME="E1:11_1_1_8"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E2:11_1_1_8"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:11_1_1_8"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E4:11_1_1_8"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</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">a'</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b'</font></span>)</span>,<font color="Maroon">c'</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">a'</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b'</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c'</font>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<font color="Maroon">a'</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c'</font></span>)</span>

<br/>.= 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a'</font>,<span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c'</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#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a'</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
;<br/><b>hence </b><a NAME="E7:11_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/><b>assume </b><a NAME="E1:11_1_1_9"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> 
 ;<br/><a NAME="E2:11_1_1_9"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><a NAME="E5:11_1_1_9"/>
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>;<br/><b>hence </b><a NAME="E6:11_1_1_9"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  )
 <b>by </b><i>, , , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/></div>
<b>end;</b></div>

<a NAME="E11:11_1_1"/><i><font color="Green">E59</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  &amp; 0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  )
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E12:11_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="group_1.html#K2">1.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E13:11_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:11_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> 
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a><br/>


</div>
