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<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">a</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="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="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 <br/> ( <font color="Olive" title="b4">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">x</font> iff <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</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> )</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">a</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> 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="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 <br/> ( <font color="Olive" title="b3">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x</font> iff <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</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> 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">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/> ( <font color="Olive" title="b2">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</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="Olive" title="b3">x</font> iff <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b3">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</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> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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:()"><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><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 <br/> ( <font color="Olive" title="b1">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> iff <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b2">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )</span><br/><span class="kw">let </span><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:()"><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="c5">W</font> holds <br/> ( <font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> iff <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">x</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</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="c5">W</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="c5">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> iff <font color="Maroon" title="c3">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="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )</span><br/>











<span class="kw">thus </span><a NAME="E1:39"/>
( <font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> implies <font color="Maroon" title="c3">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="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</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="c3">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="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c3">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="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">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="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span><br/><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c5">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <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="Maroon" title="c7">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E21">A1</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="E2:39_1_1"/>
<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</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="E3:39_1"/>
<font color="Maroon" title="c3">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="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_3.html#D11" target="_self" title="MIDSP_3:def.11">Def11</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">thus </span><a NAME="E2:39"/>
( <font color="Maroon" title="c3">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="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_2"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c3">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="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</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="c5">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">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="c5">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <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="Maroon" title="c7">m</font></span><br/><a NAME="E1:39_2_1"/>
<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="midsp_3.html#D11" target="_self" title="MIDSP_3:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E2:39_2_1"/>
<font color="Maroon" title="c5">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <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="Maroon" title="c7">m</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="E3:39_2"/>
<font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</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>; <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>


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