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

<span class="kw">set </span><font color="Maroon" title="c1">G</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #);<br/>
<span class="kw">thus </span><a NAME="E1:11"/>
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="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">f</font></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_1.html#D4" target="_self" title="GROUP_1:def.4">GROUP_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #) is  <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">reconsider </span><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="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E2:11_1"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c7">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>;<br/>
<a NAME="E3:11_1"/>
<font color="Maroon" title="c2">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c6">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c7">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c5">A</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c7">C</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">reconsider </span><font color="Maroon" title="c2">e</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #) ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">e</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_1.html#D3" target="_self" title="GROUP_1:def.3">GROUP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #) holds <br/> ( <font color="Olive" title="b1">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp;  ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #) st <br/>( <font color="Olive" title="b1">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Olive" title="b2">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> &amp; <font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> &amp;  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #) st <br/>( <font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">h</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">e</font> = 
<font color="Maroon" title="c4">A</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c3">h</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> &amp;  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #) st <br/>( <font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> ) )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c4">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c3">h</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #) st <br/>( <font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> =  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</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="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  #) ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c5">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">g</font> = 
<font color="Maroon" title="c4">A</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c2">e</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c5">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c4">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c2">e</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
