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<b>let </b><font color="Maroon" title="c1">FT</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">g</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>;<br/><b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font>;<br/>







<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E45">A1</font></i>: 
( <font color="Maroon" title="c2">g</font> <a href="fintopo6.html#R3" title="FINTOPO6:pred.3">is_minimum_path_in</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font> &amp; <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> )
 ;<br/>

<a NAME="E2:79"/><b>then </b><i><font color="Green" title="E46">A2</font></i>: 
( <font color="Maroon" title="c2">g</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> )
 
<b>by </b><i><a class="ref" href="fintopo6.html#D8" target="_self" title="FINTOPO6:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E3:79"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">Def6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fintopo6.html#D9" target="_self" title="FINTOPO6:def.9">FINTOPO6:def 9</a><br/>

<b>let </b><font color="Maroon" title="c6">i2</font>, <font color="Maroon" title="c7">j2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font>;<br/>



<b>assume </b><a NAME="E4:79"/><i><font color="Green" title="E47">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c6">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> &amp; 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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c6">i2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j2</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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">y</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:79_1"/><i><font color="Green" title="E48">A4</font></i>: 
( <font color="Maroon" title="c6">i2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c7">j2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1"/>
( <font color="Maroon" title="c6">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font> or <font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i2</font> )
 </span><b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_1"/><i><font color="Green" title="E49">A5</font></i>: 
<font color="Maroon" title="c6">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c9">n1</font> = <font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>set </b><font color="Maroon" title="c10">n2</font> = <font color="Maroon" title="c6">i2</font>;<br/><b>reconsider </b><font color="Maroon" title="c11">n1</font> = <font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <font color="Maroon" title="c12">n2</font> = <font color="Maroon" title="c6">i2</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:79_1_1_1"/>
<font color="Maroon" title="c6">i2</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="txt" href="fintopo6.html#E1:79_1_1_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:79_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E5:79_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E6:79_1_1_1"/><b>then </b><i><font color="Green" title="E50">A6</font></i>: 
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font> ;<br/><a NAME="E8:79_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:79_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="E10:79_1_1_1"/><b>then </b><i><font color="Green" title="E51">A7</font></i>: 
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#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E11:79_1_1_1"/>
<font color="Maroon" title="c7">j2</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> <font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E12:79_1_1_1"/><b>then </b><i><font color="Green" title="E52">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n1</font> &amp; <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="fintopo6.html#E10:79_1_1_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:79_1_1_1"/><i><font color="Green" title="E53">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><b>then </b><i><font color="Green" title="E54">A10</font></i>: <font color="Maroon" title="c13">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c4">x1</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>
;<br/><i><font color="Green" title="E55">A11</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g2</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c12">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="fintopo6.html#E13:79_1_1_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
;<br/><a NAME="E16:79_1_1_1"/><i><font color="Green" title="E56">A12</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E17:79_1_1_1"/><b>then </b><i><font color="Green" title="E57">A13</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c14">k</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c11">n1</font>;<br/><a NAME="E18:79_1_1_1"/><i><font color="Green" title="E58">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_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="E1:79_1_1_1_3_1"/>
( <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> or <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> )
 </span><b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1"/>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</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="E3:79_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E59">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><a NAME="E4:79_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E60">A16</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E17:79_1_1_1"><i><font color="Green" title="E57">A13</font></i></a>, <a class="txt" href="fintopo6.html#E18:79_1_1_1"><i><font color="Green" title="E58">A14</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:79_1_1_1_3_1_1"/><i><font color="Green" title="E61">A17</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E17:79_1_1_1"><i><font color="Green" title="E57">A13</font></i></a></i>;<br/><a NAME="E6:79_1_1_1_3_1_1"/><i><font color="Green" title="E62">A18</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:79_1_1_1_3_1_1"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><font color="Maroon" title="c13">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c11">n1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c11">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E4:79_1_1_1_3_1_1"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c5">x2</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="fintopo6.html#E4:79_1_1_1_3_1_1"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="fintopo6.html#E5:79_1_1_1_3_1_1"><i><font color="Green" title="E61">A17</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
;<br/><a NAME="E8:79_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E63">A19</font></i>: 
<font color="Maroon" title="c13">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="fintopo6.html#E15:79_1_1_1"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="fintopo6.html#E16:79_1_1_1"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E9:79_1_1_1_3_1_1"/><i><font color="Green" title="E64">A20</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/><a NAME="E10:79_1_1_1_3_1_1"/><i><font color="Green" title="E65">A21</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T21" title="FINSEQ_5:th.21">FINSEQ_5:21</a></i>;<br/><a NAME="E11:79_1_1_1_3_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T36" title="FINSEQ_5:th.36">FINSEQ_5:36</a></i>;<br/><a NAME="E12:79_1_1_1_3_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E9:79_1_1_1_3_1_1"><i><font color="Green" title="E64">A20</font></i></a>, <a class="txt" href="fintopo6.html#E10:79_1_1_1_3_1_1"><i><font color="Green" title="E65">A21</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E13:79_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E66">A22</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:79_1_1_1_3_1_1"/>
<font color="Maroon" title="c13">g2</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_3_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:79_1_1_1_3_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="fintopo6.html#E15:79_1_1_1"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="fintopo6.html#E6:79_1_1_1_3_1_1"><i><font color="Green" title="E62">A18</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">FINTOPO6:def 6</a><br/>

<b>let </b><font color="Maroon" title="c15">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c16">z1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font>;<br/>



<b>assume </b><a NAME="E2:79_1_1_1_3_1_1_2"/><i><font color="Green" title="E67">A23</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g2</font> &amp; <font color="Maroon" title="c16">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )
 ;<br/>

<a NAME="E3:79_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E68">A24</font></i>: 
<font color="Maroon" title="c15">i</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c15">i</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:79_1_1_1_3_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="fintopo6.html#E2:79_1_1_1_3_1_1_2"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:79_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E3:79_1_1_1_3_1_1_2"><i><font color="Green" title="E68">A24</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:79_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">z2</font> = <font color="Maroon" title="c13">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_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="E1:79_1_1_1_3_1_1_2_1"/>
( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n2</font> or <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">n2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E69">A25</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E70">A26</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n2</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="E3:79_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E71">A27</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E69">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><i><font color="Green" title="E72">A28</font></i>: <font color="Maroon" title="c16">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_1_3_1_1_2"><i><font color="Green" title="E67">A23</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E69">A25</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E69">A25</font></i></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>
;<br/><a NAME="E5:79_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E73">A29</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E6:79_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E69">A25</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><font color="Maroon" title="c18">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E5:79_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E73">A29</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E2:79_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E70">A26</font></i></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>
;<br/><b>hence </b><a NAME="E8:79_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c13">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c16">z1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_1_3_1_1_2"><i><font color="Green" title="E67">A23</font></i></a>, <a class="txt" href="fintopo6.html#E3:79_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E71">A27</font></i></a>, <a class="txt" href="fintopo6.html#E4:79_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E72">A28</font></i></a>, <a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1_2_1_2"/><i><font color="Green" title="E69">A30</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_1_3_1_1_2_1_2"/>
<font color="Maroon" title="c17">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E15:79_1_1_1"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_1_3_1_1_2"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E3:79_1_1_1_3_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E70">A31</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E69">A30</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:79_1_1_1_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c11">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E5:79_1_1_1_3_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c11">n1</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="E6:79_1_1_1_3_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E71">A32</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="fintopo6.html#E18:79_1_1_1"><i><font color="Green" title="E58">A14</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E7:79_1_1_1_3_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E72">A33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><i><font color="Green" title="E73">A34</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> = 
<font color="Maroon" title="c12">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">n2</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E69">A30</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c17">i</font>

;<br/><i><font color="Green" title="E74">A35</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">n2</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E69">A30</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_1_3_1_1_2_1_2_3"/>
( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">n2</font> or <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n2</font> )
 </span><b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E69">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1_2_3_1"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1_2_1_2_3_1"/>
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_1_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green" title="E75">A36</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">n2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E3:79_1_1_1_3_1_1_2_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">n2</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E4:79_1_1_1_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green" title="E76">A37</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:79_1_1_1_3_1_1_2_1_2_3_1"><i><font color="Green" title="E75">A36</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:79_1_1_1_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green" title="E77">A38</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E18:79_1_1_1"><i><font color="Green" title="E58">A14</font></i></a>, <a class="txt" href="fintopo6.html#E3:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E70">A31</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><i><font color="Green" title="E78">A39</font></i>: <font color="Maroon" title="c16">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E18:79_1_1_1"><i><font color="Green" title="E58">A14</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_1_3_1_1_2"><i><font color="Green" title="E67">A23</font></i></a>, <a class="txt" href="fintopo6.html#E3:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E70">A31</font></i></a>, <a class="txt" href="fintopo6.html#E8:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E73">A34</font></i></a>, <a class="txt" href="fintopo6.html#E4:79_1_1_1_3_1_1_2_1_2_3_1"><i><font color="Green" title="E76">A37</font></i></a>, <a class="ref" href="finseq_1.html#T86" title="FINSEQ_1:th.86">FINSEQ_1:86</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="fintopo6.html#E5:79_1_1_1_3_1_1_2_1_2_3_1"><i><font color="Green" title="E77">A38</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
;<br/><i><font color="Green" title="E79">A40</font></i>: <font color="Maroon" title="c18">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E6:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E71">A32</font></i></a>, <a class="txt" href="fintopo6.html#E9:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E74">A35</font></i></a>, <a class="ref" href="finseq_1.html#T86" title="FINSEQ_1:th.86">FINSEQ_1:86</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="fintopo6.html#E7:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E72">A33</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><a NAME="E8:79_1_1_1_3_1_1_2_1_2_3_1"/><i><font color="Green" title="E80">A41</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E9:79_1_1_1_3_1_1_2_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E70">A31</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E10:79_1_1_1_3_1_1_2_1_2_3_1"/>
<font color="Maroon" title="c13">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c16">z1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="fintopo6.html#E6:79_1_1_1_3_1_1_2_1_2_3_1"><i><font color="Green" title="E78">A39</font></i></a>, <a class="txt" href="fintopo6.html#E7:79_1_1_1_3_1_1_2_1_2_3_1"><i><font color="Green" title="E79">A40</font></i></a>, <a class="txt" href="fintopo6.html#E8:79_1_1_1_3_1_1_2_1_2_3_1"><i><font color="Green" title="E80">A41</font></i></a>, <a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1_2_3_2"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1_2_1_2_3_2"/><i><font color="Green" title="E75">A42</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n2</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E76">A43</font></i>: <font color="Maroon" title="c16">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n2</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_1_3_1_1_2"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c8">y</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>
;<br/><font color="Maroon" title="c18">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E6:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E71">A32</font></i></a>, <a class="txt" href="fintopo6.html#E9:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E74">A35</font></i></a>, <a class="ref" href="finseq_1.html#T86" title="FINSEQ_1:th.86">FINSEQ_1:86</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="fintopo6.html#E7:79_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E72">A33</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c12">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1_1_1_3_1_1_2_1_2_3_2"><i><font color="Green" title="E75">A42</font></i></a>, <a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/><b>hence </b><a NAME="E4:79_1_1_1_3_1_1_2_1_2_3_2"/>
<font color="Maroon" title="c13">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c16">z1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_1_3_1_1_2_1_2_3_2"><i><font color="Green" title="E76">A43</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="E15:79_1_1_1_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:79"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="fintopo6.html#E14:79_1_1_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="fintopo6.html#E8:79_1_1_1_3_1_1"><i><font color="Green" title="E63">A19</font></i></a>, <a class="txt" href="fintopo6.html#E13:79_1_1_1_3_1_1"><i><font color="Green" title="E66">A22</font></i></a>, <a class="ref" href="fintopo6.html#D8" target="_self" title="FINTOPO6:def.8">Def8</a></i>;<br/><a NAME="E16:79_1_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E15:79_1_1_1"><i><font color="Green" title="E55">A11</font></i></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><b>hence </b><a NAME="E17:79_1_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:79_1_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_2"/>
<font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_1_3_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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E3:79_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><b>hence </b><a NAME="E4:79_1_1_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:79_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_2"/><i><font color="Green" title="E49">A44</font></i>: 
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i2</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c9">n1</font> = <font color="Maroon" title="c6">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <font color="Maroon" title="c10">n2</font> = <font color="Maroon" title="c7">j2</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:79_1_1_2"/>
<font color="Maroon" title="c7">j2</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">i2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1_1_2"><i><font color="Green" title="E49">A44</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:79_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E5:79_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i2</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#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E6:79_1_1_2"/><b>then </b><i><font color="Green" title="E50">A45</font></i>: 
<font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font> ;<br/><a NAME="E8:79_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_2"><i><font color="Green" title="E49">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:79_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</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="E10:79_1_1_2"/><b>then </b><i><font color="Green" title="E51">A46</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</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#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E11:79_1_1_2"/>
<font color="Maroon" title="c6">i2</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> <font color="Maroon" title="c6">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E12:79_1_1_2"/><b>then </b><i><font color="Green" title="E52">A47</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</font> &amp; <font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="fintopo6.html#E10:79_1_1_2"><i><font color="Green" title="E51">A46</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:79_1_1_2"/><i><font color="Green" title="E53">A48</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><b>then </b><i><font color="Green" title="E54">A49</font></i>: <font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c4">x1</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>
;<br/><i><font color="Green" title="E55">A50</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g2</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c10">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="txt" href="fintopo6.html#E13:79_1_1_2"><i><font color="Green" title="E53">A48</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
;<br/><a NAME="E16:79_1_1_2"/><i><font color="Green" title="E56">A51</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E17:79_1_1_2"/><b>then </b><i><font color="Green" title="E57">A52</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c12">k</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">n1</font>;<br/><a NAME="E18:79_1_1_2"/><i><font color="Green" title="E58">A53</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_2_3_1"/>
( <font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> or <font color="Maroon" title="c9">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> )
 </span><b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1"/>
<font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon" title="c9">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</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="E3:79_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E59">A54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><a NAME="E4:79_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E60">A55</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E17:79_1_1_2"><i><font color="Green" title="E57">A52</font></i></a>, <a class="txt" href="fintopo6.html#E18:79_1_1_2"><i><font color="Green" title="E58">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:79_1_1_2_3_1_1"/><i><font color="Green" title="E61">A56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E17:79_1_1_2"><i><font color="Green" title="E57">A52</font></i></a></i>;<br/><a NAME="E6:79_1_1_2_3_1_1"/><i><font color="Green" title="E62">A57</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:79_1_1_2_3_1_1"><i><font color="Green" title="E59">A54</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">n1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E4:79_1_1_2_3_1_1"><i><font color="Green" title="E60">A55</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c5">x2</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="txt" href="fintopo6.html#E4:79_1_1_2_3_1_1"><i><font color="Green" title="E60">A55</font></i></a>, <a class="txt" href="fintopo6.html#E5:79_1_1_2_3_1_1"><i><font color="Green" title="E61">A56</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
;<br/><a NAME="E8:79_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E63">A58</font></i>: 
<font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_2"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="fintopo6.html#E15:79_1_1_2"><i><font color="Green" title="E55">A50</font></i></a>, <a class="txt" href="fintopo6.html#E16:79_1_1_2"><i><font color="Green" title="E56">A51</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E9:79_1_1_2_3_1_1"/><i><font color="Green" title="E64">A59</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/><a NAME="E10:79_1_1_2_3_1_1"/><i><font color="Green" title="E65">A60</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T21" title="FINSEQ_5:th.21">FINSEQ_5:21</a></i>;<br/><a NAME="E11:79_1_1_2_3_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T36" title="FINSEQ_5:th.36">FINSEQ_5:36</a></i>;<br/><a NAME="E12:79_1_1_2_3_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E9:79_1_1_2_3_1_1"><i><font color="Green" title="E64">A59</font></i></a>, <a class="txt" href="fintopo6.html#E10:79_1_1_2_3_1_1"><i><font color="Green" title="E65">A60</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E13:79_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E66">A61</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:79_1_1_2_3_1_1"/>
<font color="Maroon" title="c11">g2</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2_3_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:79_1_1_2_3_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="fintopo6.html#E15:79_1_1_2"><i><font color="Green" title="E55">A50</font></i></a>, <a class="txt" href="fintopo6.html#E6:79_1_1_2_3_1_1"><i><font color="Green" title="E62">A57</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">FINTOPO6:def 6</a><br/>

<b>let </b><font color="Maroon" title="c13">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c14">z1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font>;<br/>



<b>assume </b><a NAME="E2:79_1_1_2_3_1_1_2"/><i><font color="Green" title="E67">A62</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g2</font> &amp; <font color="Maroon" title="c14">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> )
 ;<br/>

<a NAME="E3:79_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green" title="E68">A63</font></i>: 
<font color="Maroon" title="c13">i</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">i</font> = <font color="Maroon" title="c13">i</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:79_1_1_2_3_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="fintopo6.html#E2:79_1_1_2_3_1_1_2"><i><font color="Green" title="E67">A62</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:79_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">g2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E3:79_1_1_2_3_1_1_2"><i><font color="Green" title="E68">A63</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:79_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">z2</font> = <font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_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="E1:79_1_1_2_3_1_1_2_1"/>
( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n2</font> or <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">n2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1_2_1_1"/><i><font color="Green" title="E69">A64</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_2_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E70">A65</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n2</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="E3:79_1_1_2_3_1_1_2_1_1"/><i><font color="Green" title="E71">A66</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_2_3_1_1_2_1_1"><i><font color="Green" title="E69">A64</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><i><font color="Green" title="E72">A67</font></i>: <font color="Maroon" title="c14">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_2"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_2_3_1_1_2"><i><font color="Green" title="E67">A62</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_2_3_1_1_2_1_1"><i><font color="Green" title="E69">A64</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1_1_2_3_1_1_2_1_1"><i><font color="Green" title="E69">A64</font></i></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>
;<br/><a NAME="E5:79_1_1_2_3_1_1_2_1_1"/><i><font color="Green" title="E73">A68</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E6:79_1_1_2_3_1_1_2_1_1"/>
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_2"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_2_3_1_1_2_1_1"><i><font color="Green" title="E69">A64</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><font color="Maroon" title="c16">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E5:79_1_1_2_3_1_1_2_1_1"><i><font color="Green" title="E73">A68</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E2:79_1_1_2_3_1_1_2_1_1"><i><font color="Green" title="E70">A65</font></i></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>
;<br/><b>hence </b><a NAME="E8:79_1_1_2_3_1_1_2_1_1"/>
<font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c14">z1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_2_3_1_1_2"><i><font color="Green" title="E67">A62</font></i></a>, <a class="txt" href="fintopo6.html#E3:79_1_1_2_3_1_1_2_1_1"><i><font color="Green" title="E71">A66</font></i></a>, <a class="txt" href="fintopo6.html#E4:79_1_1_2_3_1_1_2_1_1"><i><font color="Green" title="E72">A67</font></i></a>, <a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1_2_1_2"/><i><font color="Green" title="E69">A69</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_2_3_1_1_2_1_2"/>
<font color="Maroon" title="c15">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E15:79_1_1_2"><i><font color="Green" title="E55">A50</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_2_3_1_1_2"><i><font color="Green" title="E67">A62</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E3:79_1_1_2_3_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E70">A70</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E69">A69</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:79_1_1_2_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E5:79_1_1_2_3_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">n1</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="E6:79_1_1_2_3_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E71">A71</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="txt" href="fintopo6.html#E18:79_1_1_2"><i><font color="Green" title="E58">A53</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E7:79_1_1_2_3_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E72">A72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><i><font color="Green" title="E73">A73</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> = 
<font color="Maroon" title="c10">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n2</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_2"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E69">A69</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c15">i</font>

;<br/><i><font color="Green" title="E74">A74</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">n2</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_2"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="fintopo6.html#E1:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E69">A69</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_2_3_1_1_2_1_2_3"/>
( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">n2</font> or <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n2</font> )
 </span><b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E69">A69</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1_2_3_1"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1_2_1_2_3_1"/>
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_2_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green" title="E75">A75</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E3:79_1_1_2_3_1_1_2_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n2</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E4:79_1_1_2_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green" title="E76">A76</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:79_1_1_2_3_1_1_2_1_2_3_1"><i><font color="Green" title="E75">A75</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:79_1_1_2_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green" title="E77">A77</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E18:79_1_1_2"><i><font color="Green" title="E58">A53</font></i></a>, <a class="txt" href="fintopo6.html#E3:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E70">A70</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><i><font color="Green" title="E78">A78</font></i>: <font color="Maroon" title="c14">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E18:79_1_1_2"><i><font color="Green" title="E58">A53</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_2_3_1_1_2"><i><font color="Green" title="E67">A62</font></i></a>, <a class="txt" href="fintopo6.html#E3:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E70">A70</font></i></a>, <a class="txt" href="fintopo6.html#E8:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E73">A73</font></i></a>, <a class="txt" href="fintopo6.html#E4:79_1_1_2_3_1_1_2_1_2_3_1"><i><font color="Green" title="E76">A76</font></i></a>, <a class="ref" href="finseq_1.html#T86" title="FINSEQ_1:th.86">FINSEQ_1:86</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="txt" href="fintopo6.html#E5:79_1_1_2_3_1_1_2_1_2_3_1"><i><font color="Green" title="E77">A77</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
;<br/><i><font color="Green" title="E79">A79</font></i>: <font color="Maroon" title="c16">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E6:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E71">A71</font></i></a>, <a class="txt" href="fintopo6.html#E9:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E74">A74</font></i></a>, <a class="ref" href="finseq_1.html#T86" title="FINSEQ_1:th.86">FINSEQ_1:86</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="txt" href="fintopo6.html#E7:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E72">A72</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><a NAME="E8:79_1_1_2_3_1_1_2_1_2_3_1"/><i><font color="Green" title="E80">A80</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E9:79_1_1_2_3_1_1_2_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E70">A70</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E10:79_1_1_2_3_1_1_2_1_2_3_1"/>
<font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c14">z1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:79"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="fintopo6.html#E6:79_1_1_2_3_1_1_2_1_2_3_1"><i><font color="Green" title="E78">A78</font></i></a>, <a class="txt" href="fintopo6.html#E7:79_1_1_2_3_1_1_2_1_2_3_1"><i><font color="Green" title="E79">A79</font></i></a>, <a class="txt" href="fintopo6.html#E8:79_1_1_2_3_1_1_2_1_2_3_1"><i><font color="Green" title="E80">A80</font></i></a>, <a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1_2_3_2"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1_2_1_2_3_2"/><i><font color="Green" title="E75">A81</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n2</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E76">A82</font></i>: <font color="Maroon" title="c14">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n2</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E13:79_1_1_2"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_2_3_1_1_2"><i><font color="Green" title="E67">A62</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j2</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></i>
;<br/><font color="Maroon" title="c16">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E6:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E71">A71</font></i></a>, <a class="txt" href="fintopo6.html#E9:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E74">A74</font></i></a>, <a class="ref" href="finseq_1.html#T86" title="FINSEQ_1:th.86">FINSEQ_1:86</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E12:79_1_1_2"><i><font color="Green" title="E52">A47</font></i></a>, <a class="txt" href="fintopo6.html#E7:79_1_1_2_3_1_1_2_1_2"><i><font color="Green" title="E72">A72</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E1:79_1_1_2_3_1_1_2_1_2_3_2"><i><font color="Green" title="E75">A81</font></i></a>, <a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c8">y</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:79_1_1_2_3_1_1_2_1_2_3_2"/>
<font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c14">z1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:79"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="fintopo6.html#E2:79_1_1_2_3_1_1_2_1_2_3_2"><i><font color="Green" title="E76">A82</font></i></a>, <a class="ref" href="fin_topo.html#D15" title="FIN_TOPO:def.15">FIN_TOPO:def 15</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="E15:79_1_1_2_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:79"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="fintopo6.html#E14:79_1_1_2"><i><font color="Green" title="E54">A49</font></i></a>, <a class="txt" href="fintopo6.html#E8:79_1_1_2_3_1_1"><i><font color="Green" title="E63">A58</font></i></a>, <a class="txt" href="fintopo6.html#E13:79_1_1_2_3_1_1"><i><font color="Green" title="E66">A61</font></i></a>, <a class="ref" href="fintopo6.html#D8" target="_self" title="FINTOPO6:def.8">Def8</a></i>;<br/><a NAME="E16:79_1_1_2_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E15:79_1_1_2"><i><font color="Green" title="E55">A50</font></i></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><b>hence </b><a NAME="E17:79_1_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:79_1_1_2"><i><font color="Green" title="E50">A45</font></i></a>, <a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_2"/>
<font color="Maroon" title="c9">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c6">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E3:79_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><b>hence </b><a NAME="E4:79_1_1_2_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:79"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:79_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


</div>
