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<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <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> ;<br/><b>let </b><font color="Maroon" title="c2">jn</font>, <font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">jn</font>
 ;<br/>

<a NAME="E2:10"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4" title="FINTOPO4:def.4">FINTOPO4:def 4</a></i>;<br/>
<div><i><font color="Green" title="E10">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">k</font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>for <font color="Olive" title="b1">j2</font>, <font color="Olive" title="b2">jn2</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> <br/> for <font color="Olive" title="b3">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">jn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Olive" title="b1">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Olive" title="b3">p2</font>,$1 holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E2:10_1"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">j2</font>, <font color="Maroon" title="c7">jn2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c8">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c7">jn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p2</font> &amp; <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c8">p2</font>,0 )
 ;<br/>

<a NAME="E2:10_1_1"/><b>then </b><i><font color="Green" title="E14">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">jn2</font> &amp; <font color="Maroon" title="c7">jn2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<a NAME="E3:10_1_1"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c8">p2</font>
 
<b>by </b><i><a class="ref" href="fintopo3.html#T47" title="FINTOPO3:th.47">FINTOPO3:47</a>, <a class="txt" href="fintopo5.html#E1:10_1_1"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/>
<a NAME="E4:10_1_1"/><i><font color="Green" title="E16">A9</font></i>: 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">jn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c7">jn2</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="fintopo5.html#E1:10_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="fintopo4.html#D3" title="FINTOPO4:def.3">FINTOPO4:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1"/>
( <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font> or <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1 or <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font> )
 </span><b>by </b><i><a class="txt" href="fintopo5.html#E4:10_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="fintopo5.html#E3:10_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="fintopo5.html#E1:10_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_1"/>
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_2"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1_2_1"/>
( <font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 or <font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_2_1_1"/><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_1_1_1_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:10_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a>, <a class="txt" href="fintopo5.html#E1:10_1_1_1_2_1_1"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E4:10_1_1_1_2_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_2_1_2"/><i><font color="Green" title="E18">A12</font></i>: 
<font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E19">A13</font></i>: 
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_1_1_1_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:10_1_1_1_2_1_2"/>
<font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_1_1_1_2_1_2"><i><font color="Green" title="E18">A12</font></i></a></i>;<br/><a NAME="E4:10_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:10_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E6:10_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">jn2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E7:10_1_1_1_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">jn2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E8:10_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E20">A14</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">jn2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">jn2</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_1_1_1_2_1_2"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E9:10_1_1_1_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">jn2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E10:10_1_1_1_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">jn2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">jn2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">jn2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E11:10_1_1_1_2_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E8:10_1_1_1_2_1_2"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_3"><b>suppose </b></a><a NAME="E1:10_1_1_1_3"/><i><font color="Green" title="E17">A15</font></i>: 
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1_3_1"/>
( <font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_3_1_1"/>
<font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_1_1_1_3"><i><font color="Green" title="E17">A15</font></i></a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/><a NAME="E3:10_1_1_1_3_1_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">jn2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:10_1_1_1_3_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_3_1_2"/><i><font color="Green" title="E18">A16</font></i>: 
<font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E19">A17</font></i>: 
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_1_1_1_3"><i><font color="Green" title="E17">A15</font></i></a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/><a NAME="E3:10_1_1_1_3_1_2"/>
<font color="Maroon" title="c7">jn2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_1_1_1_3_1_2"><i><font color="Green" title="E18">A16</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:10_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c7">jn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E5:10_1_1_1_3_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a>, <a class="txt" href="fintopo5.html#E2:10_1_1_1_3_1_2"><i><font color="Green" title="E19">A17</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:10_1"/><i><font color="Green" title="E13">A18</font></i>: 
for <font color="Olive" title="b1">i2</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_2"/><i><font color="Green" title="E14">A19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i2</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c7">j3</font>, <font color="Maroon" title="c8">jn3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c9">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E2:10_1_2"/><i><font color="Green" title="E15">A20</font></i>: 
( <font color="Maroon" title="c8">jn3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c7">j3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ;<br/>

 <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a> 
<b>by </b><i><a class="ref" href="fintopo3.html#T48" title="FINTOPO3:th.48">FINTOPO3:48</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> :  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">i2</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b2">y</font> ) </span>}</span> 

;<br/>
<b>then consider </b><font color="Maroon" title="c10">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E5:10_1_2"/><i><font color="Green" title="E16">A21</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j3</font> &amp;  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b1">y</font> ) )
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_1_2"><i><font color="Green" title="E15">A20</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E7:10_1_2"/><i><font color="Green" title="E17">A22</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c11">y</font> )
 <b>by </b><i><a class="txt" href="fintopo5.html#E5:10_1_2"><i><font color="Green" title="E16">A21</font></i></a></i>;<br/>
<a NAME="E8:10_1_2"/>
<font color="Maroon" title="c11">y</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="c1">n</font>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">iy</font> = <font color="Maroon" title="c11">y</font> as    <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>  ;<br/>
<a NAME="E10:10_1_2"/><i><font color="Green" title="E18">A23</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">jn3</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">iy</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="fintopo5.html#E7:10_1_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="txt" href="fintopo5.html#E1:10_1_2"><i><font color="Green" title="E14">A19</font></i></a>, <a class="txt" href="fintopo5.html#E2:10_1_2"><i><font color="Green" title="E15">A20</font></i></a></i>;<br/>
<a NAME="E11:10_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c11">y</font>,0
 
<b>by </b><i><a class="txt" href="fintopo5.html#E7:10_1_2"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="fintopo3.html#T47" title="FINTOPO3:th.47">FINTOPO3:47</a></i>;<br/>
<a NAME="E12:10_1_2"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">iy</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j3</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:10_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="fintopo5.html#E5:10_1_2"><i><font color="Green" title="E16">A21</font></i></a></i>;<br/>
<a NAME="E13:10_1_2"/><b>then </b><i><font color="Green" title="E19">A24</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">jn3</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">iy</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">iy</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j3</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="fintopo5.html#E10:10_1_2"><i><font color="Green" title="E18">A23</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E14:10_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">jn3</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">iy</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">iy</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j3</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">jn3</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">iy</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">iy</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T142" title="COMPLEX1:th.142">COMPLEX1:142</a></i>;<br/>
<b>hence </b><a NAME="E15:10_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">jn3</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j3</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E13:10_1_2"><i><font color="Green" title="E19">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:10_1"/>
for <font color="Olive" title="b1">i3</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i3</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="fintopo5.html#E2:10_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="fintopo5.html#E3:10_1"><i><font color="Green" title="E13">A18</font></i></a>);<br/></i><b>hence </b><a NAME="E5:10_1"/>
( <font color="Maroon" title="c3">j</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="c1">n</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">jn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="fintopo5.html#E1:10_1"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_2"/><i><font color="Green" title="E11">A25</font></i>: 
( <font color="Maroon" title="c3">j</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="c1">n</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">jn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>for <font color="Olive" title="b1">j2</font>, <font color="Olive" title="b2">jn2</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> <br/> for <font color="Olive" title="b3">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">jn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Olive" title="b1">j2</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="c1">n</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Olive" title="b1">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Olive" title="b3">p2</font>,$1;<br/><a NAME="E2:10_2"/><i><font color="Green" title="E12">A26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">j2</font>, <font color="Maroon" title="c7">jn2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c8">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:10_2_1"/><i><font color="Green" title="E13">A27</font></i>: 
( <font color="Maroon" title="c7">jn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p2</font> &amp; <font color="Maroon" title="c6">j2</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="c1">n</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>

<a NAME="E2:10_2_1"/><b>then </b><i><font color="Green" title="E14">A28</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j2</font> &amp; <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_1_1_1"/>
( <font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 or <font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1"><b>case </b></a><a NAME="E1:10_2_1_1_1_1"/><i><font color="Green" title="E15">A29</font></i>: 
<font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_1_1_1_1"/><i><font color="Green" title="E16">A30</font></i>: 
<font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_2_1"><i><font color="Green" title="E13">A27</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:10_2_1_1_1_1"/><i><font color="Green" title="E17">A31</font></i>: 
<font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_2_1_1_1_1"><i><font color="Green" title="E15">A29</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E4:10_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="fintopo5.html#E1:10_2_1_1_1_1"><i><font color="Green" title="E15">A29</font></i></a></i>;<br/><a NAME="E5:10_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A32</font></i>: 
( <font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_2_1_1_1_1"><i><font color="Green" title="E16">A30</font></i></a>, <a class="txt" href="fintopo5.html#E3:10_2_1_1_1_1"><i><font color="Green" title="E17">A31</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_1_1_1_1_1"/>
( <font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> or <font color="Maroon" title="c7">jn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> )
 </span><b>by </b><i><a class="txt" href="fintopo5.html#E5:10_2_1_1_1_1"><i><font color="Green" title="E18">A32</font></i></a>, <a class="txt" href="fintopo5.html#E3:10_2_1_1_1_1"><i><font color="Green" title="E17">A31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_1_1_1_1_1_1"/><i><font color="Green" title="E19">A33</font></i>: 
<font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font>
 ;<br/><div class="add"><a NAME="E2:10_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>hence </b><a NAME="E3:10_2_1_1_1_1_1_1"/>
( <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font> or <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1 or <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_2_1_1_1_1_1_1"><i><font color="Green" title="E19">A33</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a>, <a class="txt" href="fintopo5.html#E2:10_2_1"><i><font color="Green" title="E14">A28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">jn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_2_1_1_1_1_1_2"/>
( <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font> or <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1 or <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_2"><b>case </b></a><a NAME="E1:10_2_1_1_1_2"/><i><font color="Green" title="E15">A34</font></i>: 
<font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_1_1_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_2_1"><i><font color="Green" title="E13">A27</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:10_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E16">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">jn2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">jn2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">jn2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E4:10_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a>, <a class="txt" href="fintopo5.html#E1:10_2_1_1_1_2"><i><font color="Green" title="E15">A34</font></i></a></i>;<br/><a NAME="E5:10_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:10_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">jn2</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="c6">j2</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E3:10_2_1_1_1_2"><i><font color="Green" title="E16">A35</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:10_2_1_1_1_2"/>
( <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">jn2</font> or <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1 or <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_2_1"><i><font color="Green" title="E14">A28</font></i></a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:10_2_1"/><b>then </b>
<font color="Maroon" title="c6">j2</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"><font color="Maroon" title="c7">jn2</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font></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/>
<a NAME="E5:10_2_1"/><b>then </b>
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c8">p2</font>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="fintopo5.html#E1:10_2_1"><i><font color="Green" title="E13">A27</font></i></a>, <a class="ref" href="fintopo4.html#D3" title="FINTOPO4:def.3">FINTOPO4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:10_2_1"/>
<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c8">p2</font>,0
 <b>by </b><i><a class="ref" href="fintopo3.html#T47" title="FINTOPO3:th.47">FINTOPO3:47</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:10_2"/><i><font color="Green" title="E13">A36</font></i>: 
for <font color="Olive" title="b1">jj</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">jj</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">jj</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_2_2"/><i><font color="Green" title="E14">A37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">jj</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c7">j2</font>, <font color="Maroon" title="c8">jn2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c9">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E2:10_2_2"/><i><font color="Green" title="E15">A38</font></i>: 
( <font color="Maroon" title="c8">jn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c7">j2</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="c1">n</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">x0</font> = <font color="Maroon" title="c7">j2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<a NAME="E4:10_2_2"/><i><font color="Green" title="E16">A39</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font> &amp; <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:10_2_2"><i><font color="Green" title="E15">A38</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:10_2_2"/><i><font color="Green" title="E17">A40</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">jn2</font> &amp; <font color="Maroon" title="c8">jn2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="fintopo5.html#E2:10_2_2"><i><font color="Green" title="E15">A38</font></i></a></i>;<br/>
<div><i><font color="Green" title="E18">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_2_1_1"/>
( <font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 or <font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_1"/><i><font color="Green" title="E19">A42</font></i>: 
<font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_2_1_1_1_1"/>
( <font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="fintopo5.html#E1:10_2_2_1_1_1"><i><font color="Green" title="E19">A42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_1_1_1"/><i><font color="Green" title="E20">A43</font></i>: 
<font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E21">A44</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:10_2_2_1_1_1_1_1"/>
 <a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">filled</a>
 <b>by </b><i><a class="ref" href="fintopo4.html#T18" title="FINTOPO4:th.18">FINTOPO4:18</a></i>;<br/><a NAME="E4:10_2_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">jj</font> &amp; <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c9">p2</font> )
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_2_2_1_1_1_1_1"><i><font color="Green" title="E21">A44</font></i></a>, <a class="txt" href="fintopo5.html#E2:10_2_2"><i><font color="Green" title="E15">A38</font></i></a>, <a class="txt" href="fintopo5.html#E1:10_2_2"><i><font color="Green" title="E14">A37</font></i></a>, <a class="txt" href="fintopo5.html#E1:10_2_2_1_1_1_1_1"><i><font color="Green" title="E20">A43</font></i></a>, <a class="ref" href="fin_topo.html#D4" title="FIN_TOPO:def.4">FIN_TOPO:def 4</a></i>;<br/><b>hence </b><a NAME="E5:10_2_2_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">jj</font> &amp; <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b1">y</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_1_1_2"/><i><font color="Green" title="E20">A45</font></i>: 
<font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E3:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E21">A46</font></i>: 
<font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_2_2_1_1_1_1_2"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E22">A47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:10_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E23">A48</font></i>: 
<font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E7:10_2_2_1_1_1_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_2_2_1_1_1_1_2"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E8:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E24">A49</font></i>: 
<font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E6:10_2_2_1_1_1_1_2"><i><font color="Green" title="E23">A48</font></i></a>, <a class="txt" href="fintopo5.html#E2:10_2_2"><i><font color="Green" title="E15">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:10_2_2_1_1_1_1_2"/><i><font color="Green" title="E25">A50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E3:10_2_2_1_1_1_1_2"><i><font color="Green" title="E21">A46</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E10:10_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">jn2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E11:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E26">A51</font></i>: 
<font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E8:10_2_2_1_1_1_1_2"><i><font color="Green" title="E24">A49</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:10_2_2_1_1_1_1_2"/><i><font color="Green" title="E27">A52</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E5:10_2_2"><i><font color="Green" title="E17">A40</font></i></a>, <a class="txt" href="fintopo5.html#E4:10_2_2_1_1_1_1_2"><i><font color="Green" title="E22">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:10_2_2_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="fintopo5.html#E4:10_2_2"><i><font color="Green" title="E16">A39</font></i></a></i>;<br/><a NAME="E14:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E28">A53</font></i>: 
<font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c1">n</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E12:10_2_2_1_1_1_1_2"><i><font color="Green" title="E27">A52</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">yj2</font> = <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/><a NAME="E16:10_2_2_1_1_1_1_2"/><i><font color="Green" title="E29">A54</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E11:10_2_2_1_1_1_1_2"><i><font color="Green" title="E26">A51</font></i></a>, <a class="txt" href="fintopo5.html#E9:10_2_2_1_1_1_1_2"><i><font color="Green" title="E25">A50</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E17:10_2_2_1_1_1_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E18:10_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c11">yj2</font>,0
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_2"><i><font color="Green" title="E12">A26</font></i></a>, <a class="txt" href="fintopo5.html#E2:10_2_2"><i><font color="Green" title="E15">A38</font></i></a></i>;<br/><a NAME="E19:10_2_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">yj2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">jj</font> &amp; <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c11">yj2</font> )
 <b>by </b><i><a class="txt" href="fintopo5.html#E16:10_2_2_1_1_1_1_2"><i><font color="Green" title="E29">A54</font></i></a>, <a class="txt" href="fintopo5.html#E1:10_2_2"><i><font color="Green" title="E14">A37</font></i></a>, <a class="txt" href="fintopo5.html#E2:10_2_2"><i><font color="Green" title="E15">A38</font></i></a>, <a class="txt" href="fintopo5.html#E14:10_2_2_1_1_1_1_2"><i><font color="Green" title="E28">A53</font></i></a>, <a class="ref" href="fintopo3.html#T47" title="FINTOPO3:th.47">FINTOPO3:47</a></i>;<br/><b>hence </b><a NAME="E20:10_2_2_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">jj</font> &amp; <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b1">y</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_2"/>
<font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E19">A55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:10_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E20">A56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:10_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E21">A57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">jn2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E6:10_2_2_1_1_2"/>
<font color="Maroon" title="c7">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a>, <a class="txt" href="fintopo5.html#E4:10_2_2"><i><font color="Green" title="E16">A39</font></i></a></i>;<br/><a NAME="E7:10_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E22">A58</font></i>: 
<font color="Maroon" title="c7">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E8:10_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E23">A59</font></i>: 
<font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E4:10_2_2_1_1_2"><i><font color="Green" title="E20">A56</font></i></a>, <a class="txt" href="fintopo5.html#E5:10_2_2"><i><font color="Green" title="E17">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:10_2_2_1_1_2"/>
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E11:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E4:10_2_2"><i><font color="Green" title="E16">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="fintopo5.html#E7:10_2_2_1_1_2"><i><font color="Green" title="E22">A58</font></i></a></i>;<br/><a NAME="E12:10_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E24">A60</font></i>: 
<font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <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="c1">n</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E8:10_2_2_1_1_2"><i><font color="Green" title="E23">A59</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">pj21</font> = <font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="fintopo5.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/><a NAME="E14:10_2_2_1_1_2"/><i><font color="Green" title="E25">A61</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">jn2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">jn2</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E5:10_2_2_1_1_2"><i><font color="Green" title="E21">A57</font></i></a>, <a class="txt" href="fintopo5.html#E7:10_2_2_1_1_2"><i><font color="Green" title="E22">A58</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E15:10_2_2_1_1_2"/><i><font color="Green" title="E26">A62</font></i>: 
<font color="Maroon" title="c7">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">jn2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_2_2_1_1_2"><i><font color="Green" title="E19">A55</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">jn2</font></span>)</span>
<b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">jn2</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo5.html#E7:10_2_2_1_1_2"><i><font color="Green" title="E22">A58</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a>, <a class="txt" href="fintopo5.html#E15:10_2_2_1_1_2"><i><font color="Green" title="E26">A62</font></i></a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fintopo5.html#E14:10_2_2_1_1_2"><i><font color="Green" title="E25">A61</font></i></a>, <a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>
;<br/><a NAME="E17:10_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E27">A63</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">jn2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_2_2"><i><font color="Green" title="E15">A38</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="fintopo5.html#E7:10_2_2_1_1_2"><i><font color="Green" title="E22">A58</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E19:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c11">pj21</font>,0
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:10_2"><i><font color="Green" title="E12">A26</font></i></a>, <a class="txt" href="fintopo5.html#E2:10_2_2"><i><font color="Green" title="E15">A38</font></i></a></i>;<br/><a NAME="E20:10_2_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">pj21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">jj</font> &amp; <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c11">pj21</font> )
 <b>by </b><i><a class="txt" href="fintopo5.html#E17:10_2_2_1_1_2"><i><font color="Green" title="E27">A63</font></i></a>, <a class="txt" href="fintopo5.html#E1:10_2_2"><i><font color="Green" title="E14">A37</font></i></a>, <a class="txt" href="fintopo5.html#E2:10_2_2"><i><font color="Green" title="E15">A38</font></i></a>, <a class="txt" href="fintopo5.html#E12:10_2_2_1_1_2"><i><font color="Green" title="E24">A60</font></i></a>, <a class="ref" href="fintopo3.html#T47" title="FINTOPO3:th.47">FINTOPO3:47</a></i>;<br/><b>hence </b><a NAME="E21:10_2_2_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">jj</font> &amp; <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b1">y</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
 <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">jj</font></span>)</span> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a> 
<b>by </b><i><a class="ref" href="fintopo3.html#T48" title="FINTOPO3:th.48">FINTOPO3:48</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> :  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">jj</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b2">y</font> ) </span>}</span> 

;<br/>
<b>hence </b><a NAME="E8:10_2_2"/>
<font color="Maroon" title="c7">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c9">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="fintopo5.html#E6:10_2_2"><i><font color="Green" title="E18">A41</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:10_2"/>
for <font color="Olive" title="b1">ii</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">ii</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="fintopo5.html#E2:10_2"><i><font color="Green" title="E12">A26</font></i></a>, <a class="txt" href="fintopo5.html#E3:10_2"><i><font color="Green" title="E13">A36</font></i></a>);<br/></i><b>hence </b><a NAME="E5:10_2"/>
<font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:10_2"><i><font color="Green" title="E11">A25</font></i></a>, <a class="txt" href="fintopo5.html#E1:10"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
( <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">k</font> iff ( <font color="Maroon" title="c3">j</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="c1">n</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">jn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="fintopo5.html#E3:10"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>


</div>
