<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c4">I</font> be   <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">I</font> is  <a href="scm_halt.html#V3" title="SCM_HALT:attr.3">keepInt0_1</a>  implies <font color="Maroon" title="c4">I</font> is  <a href="scm_halt.html#V1" title="SCM_HALT:attr.1">InitClosed</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_2_1"/><span class="lab"><font color="Green" title="E9">A17</font></span>: 
<font color="Maroon" title="c4">I</font> is  <a href="scm_halt.html#V3" title="SCM_HALT:attr.3">keepInt0_1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">I</font> is  <a href="scm_halt.html#V1" title="SCM_HALT:attr.1">InitClosed</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">FI</font> =  <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">s</font> be   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="scm_halt.html#D1" target="_self" title="SCM_HALT:def.1">SCM_HALT:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  st <font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> holds <br/> for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_3.html#K1" title="SCMFSA_3:func.1">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">s</font> holds <br/> <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c6">s</font>,<font color="Olive" title="b2">n</font>)</span>)</span> <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="c4">I</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">p</font> be   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">p</font> implies  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_3.html#K1" title="SCMFSA_3:func.1">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">s</font> holds <br/> <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <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="c4">I</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:9_2_1"/><span class="lab"><font color="Green" title="E10">A18</font></span>: 
<font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_3.html#K1" title="SCMFSA_3:func.1">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">s</font> holds <br/> <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <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="c4">I</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_3.html#K1" title="SCMFSA_3:func.1">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">s</font> implies  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">n</font>)</span>)</span> <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="c4">I</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:9_2_1"/><span class="lab"><font color="Green" title="E11">A19</font></span>: 
 <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_3.html#K1" title="SCMFSA_3:func.1">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">n</font>)</span>)</span> <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="c4">I</font></span><br/>

<a NAME="E4:9_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">B19</font></span>: 
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">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>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="scm_halt.html#E3"><span class="lab"><font color="Green" title="E2">zSA0</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> not  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon">c<sub>1</sub></font>)</span>)</span> <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="c4">I</font>;<br/>
<span class="kw">assume </span><a NAME="E5:9_2_1"/>
 not  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">n</font>)</span>)</span> <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="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:9_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A20</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:9_2_1"/><span class="lab"><font color="Green" title="E14">A21</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font>]
 <span class="kw">and </span><br/><a NAME="E9:9_2_1"/><span class="lab"><font color="Green" title="E15">A22</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E6:9_2_1"><span class="lab"><font color="Green" title="E13">A20</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">s2</font> =  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c12">p0</font> = <font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c13">s0</font> = <font color="Maroon" title="c6">s</font> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> (<span class="p1">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">s</font> = <font color="Maroon" title="c6">s</font> as   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">s0</font> = <font color="Maroon" title="c6">s</font> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> (<span class="p1">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) as   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>
<a NAME="E13:9_2_1"/>
 not <font color="Maroon" title="c4">I</font> is  <a href="scm_halt.html#V3" title="SCM_HALT:attr.3">keepInt0_1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:9_2_1_1"/>
 <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_2.html#T56" title="SCMFSA_2:th.56">SCMFSA_2:56</a></span>;<br/>
<a NAME="E2:9_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A25</font></span>: 
 not  <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</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="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">s02</font> =  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Maroon" title="c10">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c17">iIC</font> = <span class="p1"><a href="sfmastr1.html#K1" title="SFMASTR1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="sfmastr1.html#K1" title="SFMASTR1:func.1">}</a></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="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">s0</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="scm_halt.html#D3" target="_self" title="SCM_HALT:def.3">SCM_HALT:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_3.html#K1" title="SCMFSA_3:func.1">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">s0</font> &amp;  ex <font color="Olive" title="b1">p</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> st <br/>( <font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">p</font> &amp;  not <!--HUMANRECFAILED--> for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c15">s0</font>,<font color="Olive" title="b2">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<a NAME="E3:9_2_1_1"/>
 <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</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="c14">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_2.html#T42" title="SCMFSA_2:th.42">SCMFSA_2:42</a></span>;<br/>
<a NAME="E4:9_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A27</font></span>: 
<font color="Maroon" title="c15">s0</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<a NAME="E5:9_2_1_1"/><span class="lab"><font color="Green" title="E18">B29</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_2_1"><span class="lab"><font color="Green" title="E11">A19</font></span></a>, <a class="ref" href="scm_halt.html#T3" target="_self" title="SCM_HALT:th.3">Th7</a></span>;<br/>
<a NAME="E6:9_2_1_1"/><span class="lab"><font color="Green" title="E19">A29</font></span>: 
 not  <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sf_mastr.html#T50" title="SF_MASTR:th.50">SF_MASTR:50</a></span>;<br/>
<a NAME="E7:9_2_1_1"/>
 not  <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="sfmastr1.html#K1" title="SFMASTR1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="sfmastr1.html#K1" title="SFMASTR1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:9_2_1_1"/><span class="kw">then </span>
 not  <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</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="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_3.html#K1" title="SCMFSA_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="scm_halt.html#E4"><span class="lab"><font color="Green" title="E3">diS</font></span></a>, <a class="txt" href="#E2:9_2_1_1"><span class="lab"><font color="Green" title="E16">A25</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:9_2_1_1"/>
 <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_3.html#K1" title="SCMFSA_3:func.1">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_2_1"><span class="lab"><font color="Green" title="E11">A19</font></span></a>, <a class="ref" href="funct_7.html#T89" title="FUNCT_7:th.89">FUNCT_7:89</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> st <br/>( <font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">p</font> &amp;  not <!--HUMANRECFAILED--> for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c15">s0</font>,<font color="Olive" title="b2">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E10:9_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">B31</font></span>: 
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">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>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">s0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="scm_halt.html#E3"><span class="lab"><font color="Green" title="E2">zSA0</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>) &amp;  not <!--HUMANRECFAILED--> for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Olive" title="b1">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">thus </span><a NAME="E11:9_2_1_1"/><span class="lab"><font color="Green" title="E21">A32</font></span>: 
<font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_2_1"><span class="lab"><font color="Green" title="E10">A18</font></span></a>, <a class="txt" href="#E8:9_2_1"><span class="lab"><font color="Green" title="E14">A21</font></span></a>, <a class="ref" href="funct_7.html#T89" title="FUNCT_7:th.89">FUNCT_7:89</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <!--HUMANRECFAILED--> for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Olive" title="b1">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E12:9_2_1_1"/>
 not  <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <font color="Maroon" title="c4">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:9_2_1_1_1"/>
 <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_2_1_1_1"/><span class="kw">then </span>
 <a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font> is    <a href="scmfsa_2.html#M2" title="SCMFSA_2:mode.2">FinSeq-Location</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_2.html#T5" title="SCMFSA_2:th.5">SCMFSA_2:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:9_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_2.html#T58" title="SCMFSA_2:th.58">SCMFSA_2:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:9_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A34</font></span>: 
<font color="Maroon" title="c15">s0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <font color="Maroon" title="c4">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T92" title="FUNCT_7:th.92">FUNCT_7:92</a></span>;<br/>
<a NAME="E14:9_2_1_1"/><span class="lab"><font color="Green" title="E23">A36</font></span>: 
<font color="Maroon" title="c15">s0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c4">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T92" title="FUNCT_7:th.92">FUNCT_7:92</a>, <a class="ref" href="sf_mastr.html#T50" title="SF_MASTR:th.50">SF_MASTR:50</a></span>;<br/>
<a NAME="E15:9_2_1_1"/><span class="lab"><font color="Green" title="E24">A37</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> holds <br/> <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c14">s</font>,<font color="Olive" title="b1">m</font>)</span>)</span> <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="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:9_2_1"><span class="lab"><font color="Green" title="E15">A22</font></span></a></span>;<br/>
<a NAME="E16:9_2_1_1"/><span class="lab"><font color="Green" title="E25">A38</font></span>: 
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:9_2_1_1"><span class="lab"><font color="Green" title="E23">A36</font></span></a>, <a class="txt" href="#E13:9_2_1_1"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="txt" href="#E2:9_2_1"><span class="lab"><font color="Green" title="E10">A18</font></span></a>, <a class="txt" href="#E11:9_2_1_1"><span class="lab"><font color="Green" title="E21">A32</font></span></a>, <a class="txt" href="#E10:9_2_1_1"><span class="lab"><font color="Green" title="E20">B31</font></span></a>, <a class="txt" href="#E15:9_2_1_1"><span class="lab"><font color="Green" title="E24">A37</font></span></a>, <a class="txt" href="#E4:9_2_1"><span class="lab"><font color="Green" title="E12">B19</font></span></a>, <a class="ref" href="sf_mastr.html#T65" title="SF_MASTR:th.65">SF_MASTR:65</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Maroon" title="c18">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E17:9_2_1_1"/>
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E18:9_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A39</font></span>: 
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <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="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E19:9_2_1_1"/><span class="lab"><font color="Green" title="E27">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9_2_1_1"><span class="lab"><font color="Green" title="E26">A39</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<span class="lab"><font color="Green" title="E28">A42</font></span>:  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Maroon" title="c18">k</font>) = 
 <a href="extpro_1.html#K4" title="EXTPRO_1:func.4">Following</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="extpro_1.html#T3" title="EXTPRO_1:th.3">EXTPRO_1:3</a></span>
<br/>.= 
 <a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:9_2_1_1"><span class="lab"><font color="Green" title="E25">A38</font></span></a>, <a class="txt" href="#E19:9_2_1_1"><span class="lab"><font color="Green" title="E27">A40</font></span></a>, <a class="ref" href="pboole.html#T143" title="PBOOLE:th.143">PBOOLE:143</a></span>
;<br/>
<a NAME="E21:9_2_1_1"/>
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> holds <br/> <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Olive" title="b1">m</font>)</span>)</span> <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="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9_2_1_1"><span class="lab"><font color="Green" title="E24">A37</font></span></a>, <a class="txt" href="#E14:9_2_1_1"><span class="lab"><font color="Green" title="E23">A36</font></span></a>, <a class="txt" href="#E13:9_2_1_1"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="txt" href="#E2:9_2_1"><span class="lab"><font color="Green" title="E10">A18</font></span></a>, <a class="txt" href="#E11:9_2_1_1"><span class="lab"><font color="Green" title="E21">A32</font></span></a>, <a class="txt" href="#E10:9_2_1_1"><span class="lab"><font color="Green" title="E20">B31</font></span></a>, <a class="txt" href="#E4:9_2_1"><span class="lab"><font color="Green" title="E12">B19</font></span></a>, <a class="ref" href="sf_mastr.html#T65" title="SF_MASTR:th.65">SF_MASTR:65</a></span>;<br/>
<a NAME="E22:9_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_2_1_1"><span class="lab"><font color="Green" title="E19">A29</font></span></a>, <a class="txt" href="#E4:9_2_1_1"><span class="lab"><font color="Green" title="E17">A27</font></span></a>, <a class="txt" href="#E2:9_2_1"><span class="lab"><font color="Green" title="E10">A18</font></span></a>, <a class="txt" href="#E8:9_2_1"><span class="lab"><font color="Green" title="E14">A21</font></span></a>, <a class="txt" href="#E5:9_2_1_1"><span class="lab"><font color="Green" title="E18">B29</font></span></a>, <a class="ref" href="funct_7.html#T89" title="FUNCT_7:th.89">FUNCT_7:89</a>, <a class="ref" href="sf_mastr.html#T61" title="SF_MASTR:th.61">SF_MASTR:61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:9_2_1_1"/>
 not <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">:=</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K6" title="SF_MASTR:func.6">FirstNotUsed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c15">s0</font>,<font color="Maroon" title="c18">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:9_2_1_1"><span class="lab"><font color="Green" title="E28">A42</font></span></a>, <a class="ref" href="scmfsa_2.html#T63" title="SCMFSA_2:th.63">SCMFSA_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:9_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2_1"><span class="lab"><font color="Green" title="E9">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
