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<div class="add">

<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="Red">::  thesis: </font></span></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> holds <br/> ( <font color="Olive" title="b1">RAS</font> is  <a href="midsp_3.html#V2" title="MIDSP_3:attr.2">being_invariance</a>  iff for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">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="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="b2">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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:()"><span class="comment"><font color="Red">::  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="c2">RAS</font> holds <br/> ( <font color="Maroon" title="c2">RAS</font> is  <a href="midsp_3.html#V2" title="MIDSP_3:attr.2">being_invariance</a>  iff 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="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="b1">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">RAS</font> is  <a href="midsp_3.html#V2" title="MIDSP_3:attr.2">being_invariance</a>  iff 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">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="c3">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">x</font> )</span><br/>





<a NAME="E1:42"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
( ( 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">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="c3">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">x</font> ) implies <font color="Maroon" title="c2">RAS</font> is  <a href="midsp_3.html#V2" title="MIDSP_3:attr.2">being_invariance</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
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">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="c3">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">RAS</font> is  <a href="midsp_3.html#V2" title="MIDSP_3:attr.2">being_invariance</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RAS</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="midsp_3.html#D5" target="_self" title="MIDSP_3:def.5">MIDSP_3:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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="b2">p</font>, <font color="Olive" title="b3">q</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>  st ( for <font color="Olive" title="b4">m</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b4">m</font></span>)</span> ) holds <br/><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">b</font>,<font color="Olive" title="b3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<font color="Olive" title="b2">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>  st ( for <font color="Olive" title="b3">m</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b3">m</font></span>)</span> ) holds <br/><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">b</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<font color="Olive" title="b1">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</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:()"><span class="comment"><font color="Red">::  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> holds  <font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span> ) implies <font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> )</span><br/>







<span class="lab"><font color="Green" title="E25">A3</font></span>: <font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span> = 
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span>

<br/>.= 
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c5">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">b</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span>

;<br/>
<span class="kw">assume </span><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
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> holds  <font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E27">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c4">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="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">q</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">m</font></span><br/><a NAME="E1:42_1_2"/>
<font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<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="c8">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a></span>;<br/><a NAME="E2:42_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<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="c8">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T39" title="MIDSP_2:th.39">MIDSP_2:39</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c4">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="c8">m</font> = 
<font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<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="c8">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">q</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c8">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> = 
<font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T17" target="_self" title="MIDSP_3:th.17">Th17</a></span>
<br/>.= 
<font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">b</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">q</font></span>)</span> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="midsp_3.html#T16" target="_self" title="MIDSP_3:th.16">Th16</a></span>
<br/>.= 
<font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">b</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T17" target="_self" title="MIDSP_3:th.17">Th17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:42_1"/>
<font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T39" title="MIDSP_2:th.39">MIDSP_2:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c2">RAS</font> is  <a href="midsp_3.html#V2" title="MIDSP_3:attr.2">being_invariance</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">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="c3">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  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="c3">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c4">a</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c5">b</font>,<font color="Olive" title="b1">x</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="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">x</font></span><br/><span class="kw">set </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font>;<br/><span class="lab"><font color="Green" title="E25">A8</font></span>: <font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span> = 
<font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T17" target="_self" title="MIDSP_3:th.17">Th17</a></span>
<br/>.= 
<font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c5">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T17" target="_self" title="MIDSP_3:th.17">Th17</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span><br/><font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c9">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c9">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></span>
<br/>.= 
<font color="Maroon" title="c3">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">b</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></span>
;<br/><span class="kw">hence </span><a NAME="E2:42_2_2"/>
<font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T39" title="MIDSP_2:th.39">MIDSP_2:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:42_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="midsp_3.html#D5" target="_self" title="MIDSP_3:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:42_2"/>
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T39" title="MIDSP_2:th.39">MIDSP_2:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:42"/>
( <font color="Maroon" title="c2">RAS</font> is  <a href="midsp_3.html#V2" title="MIDSP_3:attr.2">being_invariance</a>  iff 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">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="c3">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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