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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">i</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">RAS</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="midsp_1.html#V2" title="MIDSP_1:attr.2">MidSp-like</a>  <a href="midsp_3.html#L1" title="MIDSP_3:struct.1">ReperAlgebraStr</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2<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><br/> for <font color="Olive" title="b3">v</font> being  <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Olive" title="b2">W</font><br/> 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="b2">W</font> holds <br/> ( ( for <font color="Olive" title="b5">l</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> holds <br/><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> <font color="Maroon" title="c2">i</font>,<font color="Olive" title="b3">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> ) &amp; ( for <font color="Olive" title="b5">l</font>, <font color="Olive" title="b6">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b5">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">i</font> holds <br/><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> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b3">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b5">l</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">RAS</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="midsp_1.html#V2" title="MIDSP_1:attr.2">MidSp-like</a>  <a href="midsp_3.html#L1" title="MIDSP_3:struct.1">ReperAlgebraStr</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2; <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><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="Olive" title="b1">W</font><br/> 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="b1">W</font> holds <br/> ( ( for <font color="Olive" title="b4">l</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> holds <br/><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> <font color="Maroon" title="c2">i</font>,<font color="Olive" title="b2">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> ) &amp; ( for <font color="Olive" title="b4">l</font>, <font color="Olive" title="b5">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b4">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">i</font> holds <br/><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> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b2">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b4">l</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"> for <font color="Olive" title="b1">v</font> being  <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c4">W</font><br/> 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="Maroon" title="c4">W</font> holds <br/> ( ( for <font color="Olive" title="b3">l</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> holds <br/><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> <font color="Maroon" title="c2">i</font>,<font color="Olive" title="b1">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> ) &amp; ( for <font color="Olive" title="b3">l</font>, <font color="Olive" title="b4">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b3">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">i</font> holds <br/><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> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b1">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b3">l</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <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 <br/> ( ( for <font color="Olive" title="b2">l</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> holds <br/><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">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> ) &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">i</font> holds <br/><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="Olive" title="b3">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b2">l</font> ) )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">l</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> holds <br/><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> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> ) &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i</font> holds <br/><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> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">l</font> ) )</span><br/>











<b>thus </b><a NAME="E1:32"/>
for <font color="Olive" title="b1">l</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> holds <br/><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> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</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">l</font>, <font color="Olive" title="b2">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i</font> holds <br/><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> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">l</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">l</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"> ( <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies <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> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> )</span><br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ; <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="c6">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font></span><br/>

<a NAME="E2:32_1"/>
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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>
 
<b>by </b><i><a class="ref" href="midsp_3.html#T8" target="_self" title="MIDSP_3:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E3:32_1"/>
<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> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E1:32_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="midsp_3.html#E6"><i><font color="Green" title="E2">Lm1</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>

<b>thus </b><a NAME="E2:32"/>
for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i</font> holds <br/><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> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">l</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</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>
