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<span class="kw">reconsider </span><font color="Maroon" title="c2">one1</font> = 1 as    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#T24" target="_self" title="MIDSP_3:th.24">Th24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">M</font> being   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">D</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">k</font> = <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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">C</font> = <span class="p1">(<span class="default"><span class="p2">(<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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> = $1 <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">one1</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">R</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font><span class="kw"> such that </span><br/><a NAME="E4:46_1_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> holds  <font color="Maroon" title="c7">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c8">R</font> = <font color="Maroon" title="c7">R</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">RA</font> =  <a href="midsp_3.html#G1" title="MIDSP_3:aggr.1">ReperAlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font>,the <a href="midsp_1.html#U1" title="MIDSP_1:sel.1">MIDPOINT</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">R</font> #) as  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 <span class="kw">by</span> <span class="lab"><a class="txt" href="midsp_3.html#E8"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">RA</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="c9">RA</font> is  <a href="midsp_3.html#V2" title="MIDSP_3:attr.2">being_invariance</a> </span><br/>

<a NAME="E7:46_1_1"/>
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="c9">RA</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">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="c9">RA</font>  st ( for <font color="Olive" title="b5">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="Olive" title="b1">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">q</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b5">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b5">m</font></span>)</span> ) holds <br/><font color="Olive" title="b1">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="b2">b</font>,<font color="Olive" title="b4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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="Olive" title="b1">a</font>,<font color="Olive" title="b3">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c9">RA</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="c9">RA</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="c10">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="c11">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="c10">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="c11">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="c11">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="c10">a</font>,<font color="Olive" title="b1">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">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="c9">RA</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="c10">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c11">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c10">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="c11">b</font>,<font color="Maroon" title="c13">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c10">a</font>,<font color="Maroon" title="c12">p</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:46_1_1_1"/>
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="c10">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c11">b</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c10">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="c11">b</font>,<font color="Maroon" title="c13">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c10">a</font>,<font color="Maroon" title="c12">p</font></span>)</span></span><br/>

<span class="lab"><font color="Green" title="E26">A2</font></span>:  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">p</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="midsp_3.html#K2" title="MIDSP_3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c10">a</font></span><a href="midsp_3.html#K2" title="MIDSP_3:func.2">*&gt;</a></span> <a href="midsp_3.html#K3" title="MIDSP_3:func.3">^</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">one1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></span>
;<br/>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c13">q</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="midsp_3.html#K2" title="MIDSP_3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">b</font></span><a href="midsp_3.html#K2" title="MIDSP_3:func.2">*&gt;</a></span> <a href="midsp_3.html#K3" title="MIDSP_3:func.3">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">one1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:46_1_1_1"/>
<font color="Maroon" title="c10">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="c11">b</font>,<font color="Maroon" title="c13">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c10">a</font>,<font color="Maroon" title="c12">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1_1"><span class="lab"><font color="Green" title="E26">A2</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:46_1_1"/>
<font color="Maroon" title="c9">RA</font> is  <a href="midsp_3.html#V2" title="MIDSP_3:attr.2">being_invariance</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D5" target="_self" title="MIDSP_3:def.5">Def5</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>
