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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp;  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) is <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a> )</span><br/>

<a NAME="E1:34"/><i><font color="Green" title="E31">A1</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c2">Zp</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #);<br/>
<a NAME="E2:34"/><i><font color="Green" title="E32">A2</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E3:34"/>
not 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:34"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_7.html#E2:34"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:34"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">e</font> =  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span> 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>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) <b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_3.html#T24" title="INT_3:th.24">INT_3:24</a></i>;<br/>
<div><i><font color="Green" title="E33">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) st <br/>( <font color="Maroon" title="c4">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="c3">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="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/><a NAME="E1:34_1"/>
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font>
 ;<br/><a NAME="E2:34_1"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">hp</font> = <font color="Maroon" title="c4">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> = 
<font color="Maroon" title="c5">hp</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) st <br/>( <font color="Maroon" title="c4">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="c3">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="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/><b>thus </b><font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">hp</font>
<b>by </b><i><a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) st <br/>( <font color="Maroon" title="c4">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="c3">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="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/><a NAME="E6:34_1"/>
not <font color="Maroon" title="c4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="int_7.html#E2:34_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:34_1"/><b>then </b><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c5">hp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E8:34_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/><a NAME="E9:34_1"/><b>then </b>
<font color="Maroon" title="c5">hp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E7:34_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">hpd</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span><b> such that </b><br/><a NAME="E11:34_1"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c6">hpd</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">hp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a></i>;<br/><a NAME="E12:34_1"/>
<font color="Maroon" title="c6">hpd</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E11:34_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>;<br/><a NAME="E13:34_1"/><b>then </b>
<font color="Maroon" title="c6">hpd</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/><a NAME="E14:34_1"/><b>then </b>
not <font color="Maroon" title="c6">hpd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E15:34_1"/><b>then </b>
<font color="Maroon" title="c6">hpd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c6">hpd</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>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) <b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/><a NAME="E17:34_1"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 <b>by </b><i><a class="txt" href="int_7.html#E11:34_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>;<br/><a NAME="E18:34_1"/>
<font color="Maroon" title="c7">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 <b>by </b><i><a class="txt" href="int_7.html#E11:34_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>;<br/><b>hence </b><a NAME="E19:34_1"/>
 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>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) st <br/>( <font color="Maroon" title="c4">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="c3">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="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )
 <b>by </b><i><a class="txt" href="int_7.html#E17:34_1"><i><font color="Green" title="E37">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E8:34"/><i><font color="Green" title="E34">A8</font></i>: 
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</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>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span></span><br/>

<a NAME="E1:34_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font>
 
;<br/>
<a NAME="E2:34_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:34_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font>
 
;<br/>
<a NAME="E5:34_2"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:34_2"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font>
 
;<br/>
<a NAME="E8:34_2"/><b>then </b>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">z1</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:34_2"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">y1</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>;<br/>
<a NAME="E11:34_2"/><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c9">z1</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">y1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c9">z1</font>
<b>by </b><i><a class="txt" href="int_7.html#E10:34_2"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c9">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="txt" href="int_7.html#E11:34_2"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E13:34_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:34"/>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E1:34_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font>
 
;<br/>
<a NAME="E2:34_3"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:34_3"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font>
 
;<br/>
<a NAME="E5:34_3"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> = 
<font color="Maroon" title="c6">x1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c7">y1</font>
<b>by </b><i><a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="int_7.html#E33"><i><font color="Green" title="E29">Lm12</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E8:34_3"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:34"/>
(  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp;  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_7.html#K2" title="INT_7:func.2">multint0</a> <font color="Maroon" title="c1">p</font></span>)</span> #) is <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a> )
 <b>by </b><i><a class="txt" href="int_7.html#E7:34"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="int_7.html#E8:34"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
