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<span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">SAt</font> =  <a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">IAt</font> = <font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span>)</span></span>)</span></span><br/><span class="kw">assume </span><a NAME="E1:17_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a>, <a class="ref" href="scmfsa_2.html#T127" title="SCMFSA_2:th.127">SCMFSA_2:127</a></span>;<br/><a NAME="E3:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:17_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/><a NAME="E6:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/><a NAME="E7:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A3</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_1"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:17_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a></span>;<br/><a NAME="E4:17_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="scmfsa6a.html#T37" title="SCMFSA6A:th.37">SCMFSA6A:37</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="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:17_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="scmfsa6a.html#T37" title="SCMFSA6A:th.37">SCMFSA6A:37</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E2:17"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:17"/>
 <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" title="AMI_1:func.18">Start-At</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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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