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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">m</font> be    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">RAS</font> be    <a href="midsp_3.html#M2" title="MIDSP_3:mode.2">ReperAlgebra</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c4">W</font> be   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c3">RAS</font>;<br/>







<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E38">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b2">v</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:56"/>
for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font>;<br/>

<b>set </b><font color="Maroon" title="c6">v</font> = <font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font>;<br/>
<b>set </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<a NAME="E1:56_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="ref" href="midsp_3.html#T15" target="_self" title="MIDSP_3:th.15">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_2"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:56_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:56_1"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="ref" href="midsp_3.html#T16" target="_self" title="MIDSP_3:th.16">Th16</a></i>;<br/>
<b>thus </b> <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> = 
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#D1" title="MIDSP_2:def.1">MIDSP_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:56"><i><font color="Green" title="E38">A1</font></i></a></i>
<br/>.= 
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_3.html#E2:56_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="midsp_2.html#D1" title="MIDSP_2:def.1">MIDSP_2:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:56"/>
<font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="midsp_3.html#T31" target="_self" title="MIDSP_3:th.31">Th31</a></i>;<br/>


</div>
