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<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/>

<a NAME="E1:16"/><i><font color="Green">E42</font></i>: 
2 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:16_1"/><b>then </b>
( 2 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default">0,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I</font>,<span class="p5"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 8, <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">1,2,3,4</span><a href="enumset1.html#K2">}</a></span> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="tarski.html#K4">[</a><span class="default">6,<span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">7,<span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">i</font>,<font color="Olive">a</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a> , <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : verum </span>}</span>  or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">5,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">i</font>,<font color="Olive">r</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:16_1"/><b>then </b>
( 2 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">0,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I</font>,<span class="p4"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 8, <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">1,2,3,4</span><a href="enumset1.html#K2">}</a></span> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">6,<span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  : verum </span>}</span>  or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">7,<span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">i</font>,<font color="Olive">a</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a> , <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : verum </span>}</span>  or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">5,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">i</font>,<font color="Olive">r</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:16_1"/><b>then </b>
( 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I</font>,<span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 8, <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">1,2,3,4</span><a href="enumset1.html#K2">}</a></span> </span>}</span>  or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">6,<span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  : verum </span>}</span>  or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">7,<span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">i</font>,<font color="Olive">a</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a> , <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : verum </span>}</span>  or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">5,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">i</font>,<font color="Olive">r</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:16_1"/><b>then </b>
( 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I</font>,<span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 8, <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">1,2,3,4</span><a href="enumset1.html#K2">}</a></span> </span>}</span>  or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">6,<span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  : verum </span>}</span>  or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">7,<span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">i</font>,<font color="Olive">a</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a> , <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : verum </span>}</span>  or 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">5,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">i</font>,<font color="Olive">r</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:16_1"/><b>then </b>
( 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span> or  ex <font color="Olive">I</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 8 ex <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  st <br/>( 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I</font>,<span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">d1</font>,<font color="Olive">d2</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">1,2,3,4</span><a href="enumset1.html#K2">}</a></span> ) or  ex <font color="Olive">i1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  st 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">6,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">i1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> or  ex <font color="Olive">i2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  ex <font color="Olive">d3</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  st 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">7,<span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">i2</font>,<font color="Olive">d3</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> or  ex <font color="Olive">d4</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">5,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">d4</font>,<font color="Olive">r</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:16_1"/>
contradiction
 <b>by </b><i><a class="ref" href="ami_1.html#T3">AMI_1:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16"/>
 <a href="ami_2.html#K3">SCM-Instr-Loc</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmring1.html#K1" target="_self">SCM-Instr</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="ami_2.html#D3">AMI_2:def 3</a></i>;<br/>


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