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

<b>set </b><font color="Maroon" title="c1">G</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  #);<br/>
<a NAME="E1:20"/><i><font color="Green" title="E21">A1</font></i>: 
for <font color="Olive" title="b1">h</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  #)<br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B</font> holds <br/><font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">B</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T14" target="_self" title="GR_CY_1:th.14">Th14</a></i>;<br/>
<b>thus </b><a NAME="E2:20"/>
for <font color="Olive" title="b1">h</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">f</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">h</font>, <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  #);<br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c2">h</font>, <font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c4">f</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<a NAME="E2:20_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<a NAME="E3:20_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">C</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="gr_cy_1.html#E2:20_1"><i><font color="Green" title="E22">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">A</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">C</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="gr_cy_1.html#E3:20_1"><i><font color="Green" title="E23">A3</font></i></a></i>
;<br/>


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

<b>reconsider </b><font color="Maroon" title="c2">e</font> = 0 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  #) <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c3">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  #);<br/>

<b>reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">h</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">e</font> = 
<font color="Maroon" title="c4">A</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font>

;<br/>

<b>thus </b><font color="Maroon" title="c2">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">h</font> = 
0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font>

;<br/>

<b>reconsider </b><font color="Maroon" title="c5">g</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  #) <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">g</font>
;<br/>

<b>thus </b><font color="Maroon" title="c3">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">g</font> = 
<font color="Maroon" title="c4">A</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">A</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">e</font>

;<br/>

<b>thus </b><font color="Maroon" title="c5">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">h</font> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">e</font>

;<br/>


</div>
