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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">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">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="b5">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> holds   <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">x</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">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="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#K8" title="MIDSP_3:func.8">Phi</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">x</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">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">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#K8" title="MIDSP_3:func.8">Phi</a> <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Maroon" title="c4">b</font>,<font color="Olive" title="b2">x</font></span><br/><b>let </b><font color="Maroon" title="c5">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">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="c5">W</font> holds   <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">x</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="c5">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#K8" title="MIDSP_3:func.8">Phi</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">x</font></span><br/>











<a NAME="E1:44"/>
<font color="Maroon" title="c2">RAS</font> <a href="midsp_3.html#NR1" title="MIDSP_3:NR.1">is_invariance</a> 
 
<b>by </b><i><a class="ref" href="midsp_3.html#D14" target="_self" title="MIDSP_3:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E2:44"/>
 <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K8" title="MIDSP_3:func.8">Phi</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="midsp_3.html#T21" target="_self" title="MIDSP_3:th.21">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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