<?xml version="1.0"?>
<div class="add">

<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">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">a</font>, <font color="Olive" title="b3">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">RAS</font><br/> for <font color="Olive" title="b4">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="Olive" title="b1">RAS</font><br/> for <font color="Olive" title="b5">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="b6">v</font> being  <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Olive" title="b5">W</font><br/> for <font color="Olive" title="b7">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="b5">W</font>  st <font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Olive" title="b5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">v</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Olive" title="b5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp;  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> holds <br/> <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">v</font></span><br/><b>let </b><font color="Maroon" title="c2">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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RAS</font><br/> for <font color="Olive" title="b3">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="c2">RAS</font><br/> for <font color="Olive" title="b4">W</font> being  <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c2">RAS</font><br/> for <font color="Olive" title="b5">v</font> being  <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Olive" title="b4">W</font><br/> for <font color="Olive" title="b6">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="b4">W</font>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b6">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Olive" title="b4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">v</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Olive" title="b4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/> <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font></span><br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">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="c2">RAS</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="Maroon" title="c2">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>  st <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b4">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b3">v</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> holds <br/> <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font></span><br/><b>let </b><font color="Maroon" title="c5">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="c2">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">W</font> being  <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c2">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>  st <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b3">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">v</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> holds <br/> <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font></span><br/><b>let </b><font color="Maroon" title="c6">W</font> be   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c2">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="c6">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="c6">W</font>  st <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">v</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> holds <br/> <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font></span><br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c6">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="c6">W</font>  st <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">v</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> holds <br/> <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font></span><br/><b>let </b><font color="Maroon" title="c8">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="c6">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">a</font>,<font color="Maroon" title="c8">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">v</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )</span><br/>















<b>assume </b><a NAME="E1:49"/>
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">v</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</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> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font></span><br/>

<a NAME="E2:49"/><b>then </b>
 <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T39" title="MIDSP_2:th.39">MIDSP_2:39</a></i>;<br/>
<b>hence </b><a NAME="E3:49"/>
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="midsp_3.html#D15" target="_self" title="MIDSP_3:def.15">Def15</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>
