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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">RAS</font> being   <a href="midsp_3.html#M2" title="MIDSP_3:mode.2">ReperAlgebra</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b3">W</font> being  <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Olive" title="b2">RAS</font>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Olive" title="b2">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Olive" title="b1">m</font> iff for <font color="Olive" title="b4">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b3">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="b4">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Olive" title="b3">W</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">RAS</font> being   <a href="midsp_3.html#M2" title="MIDSP_3:mode.2">ReperAlgebra</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">W</font> being  <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Olive" title="b1">RAS</font>  st <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Olive" title="b1">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font> iff for <font color="Olive" title="b3">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b2">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="b3">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Olive" title="b2">W</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being  <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c3">RAS</font>  st <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font> iff for <font color="Olive" title="b2">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">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="b2">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Olive" title="b1">W</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies ( <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font> iff 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#K2" title="NAT_1:func.2">+</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font> ) )</span><br/>







<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font> iff 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#K2" title="NAT_1:func.2">+</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font> )</span><br/>

<b>thus </b><a NAME="E2:59"/>
( <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font> implies 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#K2" title="NAT_1:func.2">+</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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#K2" title="NAT_1:func.2">+</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font> ) implies <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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#K2" title="NAT_1:func.2">+</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span><br/>

<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>;<br/>
<b>set </b><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font>;<br/>
<b>set </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font>;<br/>
<b>set </b><font color="Maroon" title="c9">p'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<a NAME="E3:59_1"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T40" title="MIDSP_2:th.40">MIDSP_2:40</a></i>;<br/>
<a NAME="E4:59_1"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c6">a</font>,<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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="midsp_3.html#T35" target="_self" title="MIDSP_3:th.35">Th35</a></i>;<br/>
<a NAME="E5:59_1"/>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="txt" href="#E1:59_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="#E3:59_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="midsp_3.html#D9" target="_self" title="MIDSP_3:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E6:59_1"/>
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="#E4:59_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="midsp_3.html#T27" target="_self" title="MIDSP_3:th.27">Th27</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>

<b>thus </b><a NAME="E3:59"/>
( ( 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#K2" title="NAT_1:func.2">+</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font> ) implies <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green" title="E41">A5</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#K2" title="NAT_1:func.2">+</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E2:59_2"/>
for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font><br/> for <font color="Olive" title="b2">p</font> being  <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">RAS</font><br/> for <font color="Olive" title="b3">pm</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>  st <font color="Olive" title="b2">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">pm</font> holds <br/> <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b3">pm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">RAS</font><br/> for <font color="Olive" title="b2">pm</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>  st <font color="Olive" title="b1">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">pm</font> holds <br/> <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b2">pm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">RAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">pm</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>  st <font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pm</font> holds <br/> <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">pm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/><b>let </b><font color="Maroon" title="c7">pm</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pm</font> implies  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">pm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )</span><br/>





<b>assume </b><a NAME="E1:59_2_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pm</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">pm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/>

<b>set </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font>;<br/>
<b>set </b><font color="Maroon" title="c9">v</font> = <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font>;<br/>
<b>set </b><font color="Maroon" title="c10">x'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<a NAME="E2:59_2_1"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T39" title="MIDSP_2:th.39">MIDSP_2:39</a></i>;<br/>
<a NAME="E3:59_2_1"/><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="midsp_3.html#T34" target="_self" title="MIDSP_3:th.34">Th34</a></i>;<br/>
<a NAME="E4:59_2_1"/>
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="#E1:59_2"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E2:59_2_1"><i><font color="Green" title="E43">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:59_2_1"/>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">pm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="#E1:59_2_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="#E3:59_2_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="midsp_3.html#T26" target="_self" title="MIDSP_3:th.26">Th26</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>
<b>hence </b><a NAME="E3:59_2"/>
<font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R4" title="MIDSP_3:pred.4">is_alternative_in</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="midsp_3.html#D9" target="_self" title="MIDSP_3:def.9">Def9</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>


</div>
