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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K10" title="MIDSP_3:func.10">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K10" title="MIDSP_3:func.10">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K10" title="MIDSP_3:func.10">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K10" title="MIDSP_3:func.10">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/>







<span class="kw">assume </span><a NAME="E1:60"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K10" title="MIDSP_3:func.10">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/>

<span class="kw">thus </span><a NAME="E2:60"/>
( <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#K10" title="MIDSP_3:func.10">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K10" title="MIDSP_3:func.10">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="60_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">a</font> =  the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>;<br/>
<span class="kw">assume </span><a NAME="E1:60_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K10" title="MIDSP_3:func.10">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/>

<span class="kw">let </span><font color="Maroon" title="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">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/>

<span class="kw">set </span><font color="Maroon" title="c7">p</font> = ( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>,<font color="Maroon" title="c6">x</font>) <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">b</font> = ( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</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/>
<span class="kw">set </span><font color="Maroon" title="c9">p9</font> = <span class="p1">(<span class="default">( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>,<font color="Maroon" title="c6">x</font>) <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</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">( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>,<font color="Maroon" title="c6">x</font>) <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</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="E2:60_1"/>
( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</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>  the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T34" title="MIDSP_2:th.34">MIDSP_2:34</a></span>;<br/>
<a NAME="E3:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> ( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>,<font color="Maroon" title="c6">x</font>) <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</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">( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>,<font color="Maroon" title="c6">x</font>) <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</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> ( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="midsp_3.html#D7" target="_self" title="MIDSP_3:def.7">Def9</a></span>;<br/>
<a NAME="E4:60_1"/>
( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">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#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>,<font color="Maroon" title="c6">x</font>) <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</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">( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>,<font color="Maroon" title="c6">x</font>) <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="midsp_3.html#T32" target="_self" title="MIDSP_3:th.32">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:60_1"/>
 <a href="midsp_3.html#K10" title="MIDSP_3:func.10">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="midsp_3.html#T24" target="_self" title="MIDSP_3:th.24">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E3:60"/>
( (  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#K10" title="MIDSP_3:func.10">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_2"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
 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#K10" title="MIDSP_3:func.10">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:60_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#K6" title="MIDSP_3:func.6">+*</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="60_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K6" title="MIDSP_3:func.6">+*</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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K6" title="MIDSP_3:func.6">+*</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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K6" title="MIDSP_3:func.6">+*</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/>





<span class="kw">assume </span><a NAME="E1:60_2_1"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K6" title="MIDSP_3:func.6">+*</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/>

<span class="kw">set </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font>);<br/>
<span class="kw">set </span><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/>
<span class="kw">set </span><font color="Maroon" title="c10">x9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</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#K8" title="MIDSP_3:func.8">.</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="E2:60_2_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T33" title="MIDSP_2:th.33">MIDSP_2:33</a></span>;<br/>
<a NAME="E3:60_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A6</font></span>: 
 <a href="midsp_3.html#K10" title="MIDSP_3:func.10">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#K8" title="MIDSP_3:func.8">.</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#K8" title="MIDSP_3:func.8">.</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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_2"><span class="lab"><font color="Green" title="E41">A4</font></span></a></span>;<br/>
<a NAME="E4:60_2_1"/>
<font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</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#K6" title="MIDSP_3:func.6">+*</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#K8" title="MIDSP_3:func.8">.</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#K8" title="MIDSP_3:func.8">.</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="midsp_3.html#T31" target="_self" title="MIDSP_3:th.31">Th34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:60_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#K6" title="MIDSP_3:func.6">+*</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_2_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E3:60_2_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="midsp_3.html#T23" target="_self" title="MIDSP_3:th.23">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:60_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D7" target="_self" title="MIDSP_3:def.7">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div>
