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<b>let </b><font color="Maroon" title="c2">a</font> be    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ;<br/><b>let </b><font color="Maroon" title="c3">k</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1"/>
( ( <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) or  not <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">k1</font> being    <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> <b> such that </b><br/><a NAME="E3:53_1_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c5">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <b>and </b><br/><a NAME="E4:53_1_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa_7.html#D3" title="SCMFSA_7:def.3">SCMFSA_7:def 3</a></i>;<br/> <a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">k</font> = 
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="scmfsa7b.html#E1:53_1_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="scmfsa7b.html#E3:53_1_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="scmfsa7b.html#E4:53_1_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
;<br/><a NAME="E6:53_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E7:53_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E1:53_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E8:53_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_2"/><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">k1</font> being    <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> <b> such that </b><br/><a NAME="E3:53_1_1_2"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c5">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <b>and </b><br/><a NAME="E4:53_1_1_2"/><i><font color="Green" title="E32">A7</font></i>: 
 <a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa_7.html#D3" title="SCMFSA_7:def.3">SCMFSA_7:def 3</a></i>;<br/><a NAME="E5:53_1_1_2"/>
<font color="Maroon" title="c5">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E1:53_1_1_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="scmfsa7b.html#E3:53_1_1_2"><i><font color="Green" title="E31">A6</font></i></a></i>;<br/><a NAME="E6:53_1_1_2"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c5">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">k</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></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#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E4:53_1_1_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI: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="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E6:53_1_1_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>
;<br/><a NAME="E8:53_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">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"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c4">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="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E1:53_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:53_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E10:53_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_3"><b>suppose </b></a><a NAME="E1:53_1_1_3"/><i><font color="Green" title="E30">A9</font></i>: 
not <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">k1</font> being    <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> <b> such that </b><br/><a NAME="E3:53_1_1_3"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E4:53_1_1_3"/><i><font color="Green" title="E32">A11</font></i>: 
 <a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa_7.html#D3" title="SCMFSA_7:def.3">SCMFSA_7:def 3</a></i>;<br/><a NAME="E5:53_1_1_3"/>
<font color="Maroon" title="c5">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E1:53_1_1_3"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="scmfsa7b.html#E3:53_1_1_3"><i><font color="Green" title="E31">A10</font></i></a></i>;<br/><a NAME="E6:53_1_1_3"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c5">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">k</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></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#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E4:53_1_1_3"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI: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="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E6:53_1_1_3"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>
;<br/><a NAME="E8:53_1_1_3"/><b>then </b>
( <font color="Maroon" title="c4">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"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c4">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="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E1:53_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:53_1_1_3"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E10:53_1_1_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:53"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div>
