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

<b>set </b><font color="Maroon" title="c1">R</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  #);<br/>
<a NAME="E1:114_1_1"/>
(  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  #) is <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> &amp;  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  #) is <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> &amp;  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  #) is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:114_1_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>( <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  #)) <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  &amp; <font color="Maroon">H<sub>2</sub></font>( <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  #)) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>( <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  #)) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D10" target="_self" title="MONOID_0:def.10">MONOID_0:def 10</a>,<a class="ref" href="monoid_0.html#D11" target="_self" title="MONOID_0:def.11">MONOID_0:def 11</a>,<a class="ref" href="monoid_0.html#D12" target="_self" title="MONOID_0:def.12">MONOID_0:def 12</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:114_1_1"/>
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  #) is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> 
 ;<br/>


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