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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="euclid_7.html#NM1" title="EUCLID_7:NM.1">Orthogonal_Basis</a> of <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b1">D</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span><br/><b>let </b><font color="Maroon" title="c2">D</font> be   <a href="euclid_7.html#NM1" title="EUCLID_7:NM.1">Orthogonal_Basis</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">D</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span><br/>



<b>set </b><font color="Maroon" title="c3">V</font> =  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c2">D</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1"><b>suppose </b></a><a NAME="E1:108_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">D</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span><br/><div class="add"><b>hence </b><a NAME="E2:108_1_1"/>
<font color="Maroon" title="c2">D</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="euclid_7.html#T19" target="_self" title="EUCLID_7:th.19">Th21</a>, <a class="ref" href="euclid_7.html#T51" target="_self" title="EUCLID_7:th.51">Th57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2"><b>suppose </b></a><a NAME="E1:108_1_2"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">D</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c5">D0</font> = <font color="Maroon" title="c2">D</font> as  <a href="euclid_7.html#V3" title="EUCLID_7:attr.3">R-orthonormal</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><a NAME="E3:108_1_2"/><i><font color="Green" title="E60">A3</font></i>: 
for <font color="Olive" title="b1">D2</font> being <a href="euclid_7.html#V3" title="EUCLID_7:attr.3">R-orthonormal</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Maroon" title="c5">D0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D2</font> holds <br/><font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D0</font>
 <b>by </b><i><a class="ref" href="euclid_7.html#D6" target="_self" title="EUCLID_7:def.6">Def7</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">I</font> being    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:108_1_2"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="ref" href="rlvect_5.html#T2" title="RLVECT_5:th.2">RLVECT_5:2</a></i>;<br/><a NAME="E6:108_1_2"/>
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="txt" href="#E5:108_1_2"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="rlvect_3.html#T23" title="RLVECT_3:th.23">RLVECT_3:23</a></i>;<br/><a NAME="E7:108_1_2"/><b>then </b><i><font color="Green" title="E62">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c6">I</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/><a NAME="E8:108_1_2"/><i><font color="Green" title="E63">A6</font></i>: 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c6">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="rlvect_3.html#D3" title="RLVECT_3:def.3">RLVECT_3:def 3</a></i>;<br/><a NAME="E9:108_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="euclid_7.html#T50" target="_self" title="EUCLID_7:th.50">Th56</a></i>;<br/><a NAME="E10:108_1_2"/><b>then </b><i><font color="Green" title="E64">A7</font></i>: 
<font color="Maroon" title="c6">I</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><div><i><font color="Green" title="E65">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_1_2_1"/>
<font color="Maroon" title="c6">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">B</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span><br/><a NAME="E2:108_1_2_1"/><b>then </b>
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c6">I</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="rlvect_5.html#T20" title="RLVECT_5:th.20">RLVECT_5:20</a></i>;<br/><a NAME="E3:108_1_2_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/><a NAME="E4:108_1_2_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:108_1_2"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E5:108_1_2_1"/><b>then </b>
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T38" title="RLSUB_1:th.38">RLSUB_1:38</a></i>;<br/><b>hence </b><a NAME="E6:108_1_2_1"/>
<font color="Maroon" title="c4">B</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="#E8:108_1_2"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="rlvect_3.html#D3" title="RLVECT_3:def.3">RLVECT_3:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_1_2_2"/>
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">I</font> &amp; not <font color="Maroon" title="c6">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">B</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span><br/><b>then consider </b><font color="Maroon" title="c7">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:108_1_2_2"/><i><font color="Green" title="E66">A9</font></i>: 
( <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I</font> &amp; not <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E4:108_1_2_2"/>
not <font color="Maroon" title="c7">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2"><i><font color="Green" title="E66">A9</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E5:108_1_2_2"/><b>then </b><i><font color="Green" title="E67">A10</font></i>: 
( <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I</font> &amp; not <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2"><i><font color="Green" title="E66">A9</font></i></a>, <a class="ref" href="rlvect_3.html#T18" title="RLVECT_3:th.18">RLVECT_3:18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">z0</font> = <font color="Maroon" title="c7">x0</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="#E3:108_1_2_2"><i><font color="Green" title="E66">A9</font></i></a></i>;<br/><a NAME="E7:108_1_2_2"/><i><font color="Green" title="E68">A11</font></i>: 
<font color="Maroon" title="c6">I</font> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="rlvect_3.html#D3" title="RLVECT_3:def.3">RLVECT_3:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">B1</font> = <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">z0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E10:108_1_2_2"/><i><font color="Green" title="E69">A12</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c10">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="#E5:108_1_2"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="#E10:108_1_2"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/><a NAME="E11:108_1_2_2"/>
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="c10">p</font>
 <b>by </b><i><a class="txt" href="#E1:108_1_2"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="finseq_3.html#T34" title="FINSEQ_3:th.34">FINSEQ_3:34</a></i>;<br/><a NAME="E12:108_1_2_2"/><b>then </b><i><font color="Green" title="E70">A13</font></i>: 
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="c10">p</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">p0</font> = <font color="Maroon" title="c10">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">a00</font> = <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c13">z00</font> = <font color="Maroon" title="c8">z0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c8">z0</font>,<font color="Maroon" title="c12">a00</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">a00</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><b>set </b><font color="Maroon" title="c14">z01</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c13">z00</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c13">z00</font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  ex <font color="Olive" title="b1">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( $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="c10">p</font> implies for <font color="Olive" title="b2">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds not <font color="Olive" title="b1">q</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="Olive" title="b2">d</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> ) &amp; <font color="Olive" title="b1">q</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="c8">z0</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> holds <br/>( <font color="Olive" title="b1">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> ) ) ) );<br/><a NAME="E16:108_1_2_2"/><i><font color="Green" title="E71">A14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E17:108_1_2_2"/><i><font color="Green" title="E72">A15</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:108_1_2_2_1"/><i><font color="Green" title="E73">A16</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_2_1_1"/>
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">k</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  not <font color="Maroon" title="c15">k</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_1_1"/><i><font color="Green" title="E74">A17</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">k</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><a NAME="E2:108_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E75">A18</font></i>: 
<font color="Maroon" title="c15">k</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="c10">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:108_1_2_2_1_1_1"/>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E4:108_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E76">A19</font></i>: 
<font color="Maroon" title="c15">k</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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1"><i><font color="Green" title="E74">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_2_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> or 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">q0</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:108_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E77">A20</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c15">k</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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( <font color="Maroon" title="c15">k</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="c10">p</font> implies for <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds not <font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span> ) &amp; <font color="Maroon" title="c16">q0</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="c8">z0</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> holds <br/>( <font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_1"><i><font color="Green" title="E73">A16</font></i></a>, <a class="txt" href="#E4:108_1_2_2_1_1_1"><i><font color="Green" title="E76">A19</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">a2</font> = <font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font>, <font color="Maroon" title="c18">b2</font> = <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c19">q3</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c17">a2</font>,<font color="Maroon" title="c18">b2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c20">q1</font> = <font color="Maroon" title="c16">q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">q3</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><i><font color="Green" title="E78">A21</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">q1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q0</font></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> <font color="Maroon" title="c19">q3</font></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="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/><a NAME="E8:108_1_2_2_1_1_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:108_1_2_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E79">A22</font></i>: 
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="c16">q0</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:108_1_2_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E80">A23</font></i>: 
<font color="Maroon" title="c20">q1</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="c8">z0</font>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E11:108_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E81">A24</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <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="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  not <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> );<br/><a NAME="E12:108_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E82">A25</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2"><i><font color="Green" title="E66">A9</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E9:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E79">A22</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E13:108_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E83">A26</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">j</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_2"/><i><font color="Green" title="E84">A27</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:108_1_2_2_1_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  not <font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/>

<a NAME="E2:108_1_2_2_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E85">A28</font></i>: 
<font color="Maroon" title="c21">j</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="c15">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:108_1_2_2_1_1_1_1_1_2_1"/><i><font color="Green" title="E86">A29</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/>

<a NAME="E2:108_1_2_2_1_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c22">z</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#T18" title="RLVECT_3:th.18">RLVECT_3:18</a></i>;<br/>
<b>hence </b><a NAME="E3:108_1_2_2_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_2_1_1_1_1_1_2_1_2"/>
( <font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> or <font color="Maroon" title="c21">j</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="c15">k</font> )
 </span><b>by </b><i><a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_2_1"><i><font color="Green" title="E85">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_1_1_1_1_2_1_2_1"/><i><font color="Green" title="E87">A30</font></i>: 
<font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/><div class="add"><a NAME="E2:108_1_2_2_1_1_1_1_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E88">A31</font></i>: 
<font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c17">a2</font>,<font color="Maroon" title="c18">b2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/><a NAME="E3:108_1_2_2_1_1_1_1_1_2_1_2_1"/><i><font color="Green" title="E89">A32</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j</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="c15">k</font> )
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E87">A30</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E4:108_1_2_2_1_1_1_1_1_2_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="E5:108_1_2_2_1_1_1_1_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E90">A33</font></i>: 
<font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="#E7:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E78">A21</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E87">A30</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E6:108_1_2_2_1_1_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p0</font>
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_1_1_1"><i><font color="Green" title="E75">A18</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E87">A30</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E89">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:108_1_2_2_1_1_1_1_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E91">A34</font></i>: 
<font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:108_1_2_2_1_1_1_1_1_2_1_2_1"/><i><font color="Green" title="E92">A35</font></i>: 
<font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_1_1_1"><i><font color="Green" title="E75">A18</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E87">A30</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_2_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_2_1_2_1_1"/><i><font color="Green" title="E93">A36</font></i>: 
<font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:108_1_2_2_1_1_1_1_1_2_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c17">a2</font>,<font color="Maroon" title="c18">b2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E5:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E90">A33</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E87">A30</font></i></a>, <a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E88">A31</font></i></a>, <a class="ref" href="rvsum_1.html#T64" title="RVSUM_1:th.64">RVSUM_1:64</a></i>;<br/><a NAME="E3:108_1_2_2_1_1_1_1_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_2_1"><i><font color="Green" title="E86">A29</font></i></a>, <a class="txt" href="#E5:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E90">A33</font></i></a>, <a class="txt" href="#E7:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E91">A34</font></i></a>, <a class="txt" href="#E8:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E92">A35</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_1_1"><i><font color="Green" title="E93">A36</font></i></a>, <a class="txt" href="euclid_7.html#E56"><i><font color="Green" title="E46">Lm2</font></i></a></i>;<br/><a NAME="E4:108_1_2_2_1_1_1_1_1_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E94">A37</font></i>: 
<font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E5:108_1_2_2_1_1_1_1_1_2_1_2_1_1"/><i><font color="Green" title="E95">A38</font></i>: 
not <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2"><i><font color="Green" title="E84">A27</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E87">A30</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E6:108_1_2_2_1_1_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E87">A30</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><b>hence </b><a NAME="E7:108_1_2_2_1_1_1_1_1_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_1"><i><font color="Green" title="E87">A30</font></i></a>, <a class="txt" href="#E4:108_1_2_2_1_1_1_1_1_2_1_2_1_1"><i><font color="Green" title="E94">A37</font></i></a>, <a class="txt" href="#E5:108_1_2_2_1_1_1_1_1_2_1_2_1_1"><i><font color="Green" title="E95">A38</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E10:108_1_2_2_1_1_1_1_1_2_1_2_1"/>
not <font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_1_1_1_1_2_1_2_2"/><i><font color="Green" title="E87">A39</font></i>: 
<font color="Maroon" title="c21">j</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="c15">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/><div class="add"><a NAME="E2:108_1_2_2_1_1_1_1_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E88">A40</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j</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="c15">k</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">a11</font> = <font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c23">b11</font> = <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E4:108_1_2_2_1_1_1_1_1_2_1_2_2"/>
( <font color="Maroon" title="c22">a11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">b11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ;<br/><a NAME="E5:108_1_2_2_1_1_1_1_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E89">A41</font></i>: 
( <font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c16">q0</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c22">a11</font>,<font color="Maroon" title="c23">b11</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E88">A40</font></i></a></i>;<br/><a NAME="E6:108_1_2_2_1_1_1_1_1_2_1_2_2"/><i><font color="Green" title="E90">A42</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="E7:108_1_2_2_1_1_1_1_1_2_1_2_2"/><i><font color="Green" title="E91">A43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E87">A39</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:108_1_2_2_1_1_1_1_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E92">A44</font></i>: 
<font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c16">q0</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E6:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E90">A42</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E9:108_1_2_2_1_1_1_1_1_2_1_2_2"/><i><font color="Green" title="E93">A45</font></i>: 
<font color="Maroon" title="c21">j</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">p0</font>
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_1_1_1"><i><font color="Green" title="E75">A18</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E87">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:108_1_2_2_1_1_1_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p0</font>
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E88">A40</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:108_1_2_2_1_1_1_1_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E94">A46</font></i>: 
<font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:108_1_2_2_1_1_1_1_1_2_1_2_2"/><i><font color="Green" title="E95">A47</font></i>: 
<font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E9:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E93">A45</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E13:108_1_2_2_1_1_1_1_1_2_1_2_2"/><i><font color="Green" title="E96">A48</font></i>: 
<font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c16">q0</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E6:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E90">A42</font></i></a>, <a class="txt" href="#E7:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E91">A43</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_2_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_2_1_2_2_1"/><i><font color="Green" title="E97">A49</font></i>: 
<font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:108_1_2_2_1_1_1_1_1_2_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c22">a11</font>,<font color="Maroon" title="c23">b11</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E5:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E89">A41</font></i></a>, <a class="txt" href="#E8:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E92">A44</font></i></a>, <a class="ref" href="rvsum_1.html#T64" title="RVSUM_1:th.64">RVSUM_1:64</a></i>;<br/><a NAME="E3:108_1_2_2_1_1_1_1_1_2_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_2_1"><i><font color="Green" title="E86">A29</font></i></a>, <a class="txt" href="#E8:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E92">A44</font></i></a>, <a class="txt" href="#E11:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E94">A46</font></i></a>, <a class="txt" href="#E12:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E95">A47</font></i></a>, <a class="txt" href="#E13:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E96">A48</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_2_1"><i><font color="Green" title="E97">A49</font></i></a>, <a class="txt" href="euclid_7.html#E56"><i><font color="Green" title="E46">Lm2</font></i></a></i>;<br/><a NAME="E4:108_1_2_2_1_1_1_1_1_2_1_2_2_1"/><b>then </b><i><font color="Green" title="E98">A50</font></i>: 
<font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E5:108_1_2_2_1_1_1_1_1_2_1_2_2_1"/>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E6:108_1_2_2_1_1_1_1_1_2_1_2_2_1"/><b>then </b><i><font color="Green" title="E99">A51</font></i>: 
not <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2"><i><font color="Green" title="E84">A27</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E87">A39</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:108_1_2_2_1_1_1_1_1_2_1_2_2_1"/><i><font color="Green" title="E100">A52</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E8:108_1_2_2_1_1_1_1_1_2_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E87">A39</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><b>hence </b><a NAME="E9:108_1_2_2_1_1_1_1_1_2_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_2_1_2_2"><i><font color="Green" title="E87">A39</font></i></a>, <a class="txt" href="#E4:108_1_2_2_1_1_1_1_1_2_1_2_2_1"><i><font color="Green" title="E98">A50</font></i></a>, <a class="txt" href="#E6:108_1_2_2_1_1_1_1_1_2_1_2_2_1"><i><font color="Green" title="E99">A51</font></i></a>, <a class="txt" href="#E7:108_1_2_2_1_1_1_1_1_2_1_2_2_1"><i><font color="Green" title="E100">A52</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E15:108_1_2_2_1_1_1_1_1_2_1_2_2"/>
not <font color="Maroon" title="c20">q1</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="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:108_1_2_2_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E14:108_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E84">A53</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E12:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E82">A25</font></i></a>, <a class="txt" href="#E13:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E83">A26</font></i></a>);<br/></i><a NAME="E15:108_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E85">A54</font></i>: 
for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b2">a01</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Olive" title="b2">a01</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">s1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a01</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Olive" title="b1">a01</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c22">a01</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c22">a01</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_3"/><i><font color="Green" title="E86">A55</font></i>: 
<font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">a01</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/>

<a NAME="E2:108_1_2_2_1_1_1_1_1_3"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">s1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">s1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )</span><br/>

<b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">s1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font></span><br/>

<b>hence </b><a NAME="E2:108_1_2_2_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_3"><i><font color="Green" title="E86">A55</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:108_1_2_2_1_1_1_1_1_3"/><b>then </b>
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">s1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#T23" title="RLVECT_3:th.23">RLVECT_3:23</a></i>;<br/>
<a NAME="E4:108_1_2_2_1_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E87">A56</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">s1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/>
<a NAME="E5:108_1_2_2_1_1_1_1_1_3"/>
<font color="Maroon" title="c22">a01</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">s1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">s1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_4.html#T11" title="RLVECT_4:th.11">RLVECT_4:11</a></i>;<br/>
<a NAME="E6:108_1_2_2_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c22">a01</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">s1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:108_1_2_2_1_1_1_1_1_3"/>
<font color="Maroon" title="c22">a01</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:108_1_2_2_1_1_1_1_1_3"><i><font color="Green" title="E87">A56</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E16:108_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E86">A57</font></i>: 
for <font color="Olive" title="b1">s1</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">a01</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Olive" title="b2">a01</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">s1</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a01</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Olive" title="b1">a01</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c22">a01</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c22">a01</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_4"/><i><font color="Green" title="E87">A58</font></i>: 
<font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">a01</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c23">s10</font> = <font color="Maroon" title="c21">s1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E3:108_1_2_2_1_1_1_1_1_4"/>
<font color="Maroon" title="c22">a01</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a01</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c23">s10</font>
 
;<br/>
<b>hence </b><a NAME="E4:108_1_2_2_1_1_1_1_1_4"/>
<font color="Maroon" title="c22">a01</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E15:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E85">A54</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_4"><i><font color="Green" title="E87">A58</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E17:108_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E87">A59</font></i>: 
( <font color="Maroon" title="c15">k</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="c10">p</font> implies <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</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> <font color="Maroon" title="c4">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_5"/><i><font color="Green" title="E88">A60</font></i>: 
<font color="Maroon" title="c15">k</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="c10">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</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> <font color="Maroon" title="c4">B</font></span><br/>

<a NAME="E2:108_1_2_2_1_1_1_1_1_5"/><b>then </b>
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">p</font>
 
<b>by </b><i><a class="txt" href="#E11:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E81">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="E3:108_1_2_2_1_1_1_1_1_5"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:108_1_2_2_1_1_1_1_1_5"/>
<font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</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> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_5"><i><font color="Green" title="E88">A60</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E18:108_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E88">A61</font></i>: 
( <font color="Maroon" title="c15">k</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="c10">p</font> implies for <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds not <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E14:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E84">A53</font></i></a>, <a class="txt" href="#E16:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E86">A57</font></i></a>, <a class="txt" href="#E17:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E87">A59</font></i></a></i>;<br/><a NAME="E19:108_1_2_2_1_1_1_1_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> holds <br/>( <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> holds <br/>( <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c22">a</font>, <font color="Maroon" title="c23">b</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> implies ( <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span></span>)</span> ) )</span><br/>



<b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_6"/><i><font color="Green" title="E89">A62</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E2:108_1_2_2_1_1_1_1_1_6"/><b>then </b><i><font color="Green" title="E90">A63</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</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="E3:108_1_2_2_1_1_1_1_1_6"/><i><font color="Green" title="E91">A64</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E4:108_1_2_2_1_1_1_1_1_6"/><i><font color="Green" title="E92">A65</font></i>: 
<font color="Maroon" title="c21">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> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E89">A62</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_2_1_1_1_1_1_6_1"/>
( <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> or <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> )
 </span><b>by </b><i><a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E90">A63</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_1_1_1_1_6_1_1"/><i><font color="Green" title="E93">A66</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span></span>)</span> )</span><br/><div class="add"><a NAME="E2:108_1_2_2_1_1_1_1_1_6_1_1"/><b>then </b><i><font color="Green" title="E94">A67</font></i>: 
<font color="Maroon" title="c21">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> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E95">A68</font></i>: <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> = 
<font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
<b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E89">A62</font></i></a>, <a class="txt" href="#E7:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E78">A21</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
<b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E89">A62</font></i></a>, <a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E90">A63</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="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font>
<b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E89">A62</font></i></a>, <a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E90">A63</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/><a NAME="E4:108_1_2_2_1_1_1_1_1_6_1_1"/><i><font color="Green" title="E96">A69</font></i>: 
<font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_6_1_1"><i><font color="Green" title="E94">A67</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E91">A64</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:108_1_2_2_1_1_1_1_1_6_1_1"/><i><font color="Green" title="E97">A70</font></i>: 
<font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E91">A64</font></i></a>, <a class="txt" href="#E4:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E92">A65</font></i></a>, <a class="txt" href="#E7:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E78">A21</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E6:108_1_2_2_1_1_1_1_1_6_1_1"/>
<font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_6_1_1"><i><font color="Green" title="E94">A67</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E91">A64</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>hence </b><a NAME="E7:108_1_2_2_1_1_1_1_1_6_1_1"/>
( <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_6_1_1"><i><font color="Green" title="E93">A66</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E89">A62</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_6_1_1"><i><font color="Green" title="E95">A68</font></i></a>, <a class="txt" href="#E4:108_1_2_2_1_1_1_1_1_6_1_1"><i><font color="Green" title="E96">A69</font></i></a>, <a class="txt" href="#E5:108_1_2_2_1_1_1_1_1_6_1_1"><i><font color="Green" title="E97">A70</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_1_1_1_1_6_1_2"/><i><font color="Green" title="E93">A71</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span></span>)</span> )</span><br/><div class="add"><a NAME="E2:108_1_2_2_1_1_1_1_1_6_1_2"/><i><font color="Green" title="E94">A72</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E3:108_1_2_2_1_1_1_1_1_6_1_2"/><i><font color="Green" title="E95">A73</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E4:108_1_2_2_1_1_1_1_1_6_1_2"/><i><font color="Green" title="E96">A74</font></i>: 
<font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:108_1_2_2_1_1_1_1_1_6_1_2"/><i><font color="Green" title="E97">A75</font></i>: 
<font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E6:108_1_2_2_1_1_1_1_1_6_1_2"/><i><font color="Green" title="E98">A76</font></i>: 
<font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E7:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E78">A21</font></i></a>, <a class="txt" href="#E2:108_1_2_2_1_1_1_1_1_6_1_2"><i><font color="Green" title="E94">A72</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><i><font color="Green" title="E99">A77</font></i>: <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E7:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E78">A21</font></i></a>, <a class="txt" href="#E11:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E81">A24</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_6_1_2"><i><font color="Green" title="E93">A71</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon" title="c19">q3</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">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E78">A21</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_6_1_2"><i><font color="Green" title="E95">A73</font></i></a>, <a class="ref" href="finseq_1.html#T37" title="FINSEQ_1:th.37">FINSEQ_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c17">a2</font>,<font color="Maroon" title="c18">b2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_1_6_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_1_6_1_2_2"/>
<font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:108_1_2_2_1_1_1_1_1_6_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c17">a2</font>,<font color="Maroon" title="c18">b2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c17">a2</font>,<font color="Maroon" title="c18">b2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c17">a2</font>,<font color="Maroon" title="c18">b2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:108_1_2_2_1_1_1_1_1_6_1_2"><i><font color="Green" title="E99">A77</font></i></a>, <a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></i>;<br/><a NAME="E3:108_1_2_2_1_1_1_1_1_6_1_2_2"/><b>then </b><i><font color="Green" title="E100">A78</font></i>: 
<font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c17">a2</font>,<font color="Maroon" title="c18">b2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T64" title="RVSUM_1:th.64">RVSUM_1:64</a></i>;<br/><a NAME="E4:108_1_2_2_1_1_1_1_1_6_1_2_2"/>
<font color="Maroon" title="c16">q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>hence </b><a NAME="E5:108_1_2_2_1_1_1_1_1_6_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1"><i><font color="Green" title="E74">A17</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_6_1_2_2"><i><font color="Green" title="E100">A78</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E9:108_1_2_2_1_1_1_1_1_6_1_2"/>
( <font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c20">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E7:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E78">A21</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E89">A62</font></i></a>, <a class="txt" href="#E3:108_1_2_2_1_1_1_1_1_6"><i><font color="Green" title="E91">A64</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_1_6_1_2"><i><font color="Green" title="E93">A71</font></i></a>, <a class="txt" href="#E4:108_1_2_2_1_1_1_1_1_6_1_2"><i><font color="Green" title="E96">A74</font></i></a>, <a class="txt" href="#E5:108_1_2_2_1_1_1_1_1_6_1_2"><i><font color="Green" title="E97">A75</font></i></a>, <a class="txt" href="#E6:108_1_2_2_1_1_1_1_1_6_1_2"><i><font color="Green" title="E98">A76</font></i></a>, <a class="txt" href="#E7:108_1_2_2_1_1_1_1_1_6_1_2"><i><font color="Green" title="E99">A77</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E20:108_1_2_2_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="#E7:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E78">A21</font></i></a>, <a class="txt" href="#E10:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E80">A23</font></i></a>, <a class="txt" href="#E18:108_1_2_2_1_1_1_1_1"><i><font color="Green" title="E88">A61</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_1_1_1_2"/>
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><a NAME="E2:108_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E77">A79</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">q1</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">z0</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E4:108_1_2_2_1_1_1_1_2"/><i><font color="Green" title="E78">A80</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c16">q1</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="c8">z0</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:108_1_2_2_1_1_1_1_2"/><i><font color="Green" title="E79">A81</font></i>: 
( 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="c10">p</font> implies for <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds not <font color="Maroon" title="c16">q1</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="Olive" title="b1">d</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_1_2_2_1_1_1_1_2_1"/><i><font color="Green" title="E80">A82</font></i>: 
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="c10">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds not <font color="Maroon" title="c16">q1</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="Olive" title="b1">d</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><br/>

<b>thus </b><a NAME="E2:108_1_2_2_1_1_1_1_2_1"/>
for <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds not <font color="Maroon" title="c16">q1</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="Olive" title="b1">d</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">d</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c16">q1</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="c17">d</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><br/>

<a NAME="E1:108_1_2_2_1_1_1_1_2_1_1"/><i><font color="Green" title="E81">A83</font></i>: 
<font color="Maroon" title="c16">q1</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="c8">z0</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E2:108_1_2_2_1_1_1_1_2_1_1"/><i><font color="Green" title="E82">A84</font></i>: 
<font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_2_1"><i><font color="Green" title="E80">A82</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E3:108_1_2_2_1_1_1_1_2_1_1"/>
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">p0</font>
 
<b>by </b><i><a class="txt" href="#E1:108_1_2_2_1_1_1_1_2_1"><i><font color="Green" title="E80">A82</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:108_1_2_2_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:108_1_2_2_1_1_1_1_2_1_1"/>
not <font color="Maroon" title="c16">q1</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="c17">d</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E5:108_1_2"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="#E5:108_1_2_2"><i><font color="Green" title="E67">A10</font></i></a>, <a class="txt" href="#E7:108_1_2_2"><i><font color="Green" title="E68">A11</font></i></a>, <a class="txt" href="#E1:108_1_2_2_1_1_1_1_2_1_1"><i><font color="Green" title="E81">A83</font></i></a>, <a class="txt" href="#E2:108_1_2_2_1_1_1_1_2_1_1"><i><font color="Green" title="E82">A84</font></i></a>, <a class="ref" href="euclid_7.html#T43" target="_self" title="EUCLID_7:th.43">Th49</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div><a NAME="E6:108_1_2_2_1_1_1_1_2"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> holds <br/>( <font color="Maroon" title="c16">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c16">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )
 ;<br/><b>hence </b><a NAME="E7:108_1_2_2_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_1_1_1_1_2"><i><font color="Green" title="E77">A79</font></i></a>, <a class="txt" href="#E4:108_1_2_2_1_1_1_1_2"><i><font color="Green" title="E78">A80</font></i></a>, <a class="txt" href="#E5:108_1_2_2_1_1_1_1_2"><i><font color="Green" title="E79">A81</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_1_2"/>
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  not <font color="Maroon" title="c15">k</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><b>hence </b><a NAME="E2:108_1_2_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E18:108_1_2_2"/><i><font color="Green" title="E73">A85</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E16:108_1_2_2"><i><font color="Green" title="E71">A14</font></i></a>, <a class="txt" href="#E17:108_1_2_2"><i><font color="Green" title="E72">A15</font></i></a>);<br/></i><a NAME="E19:108_1_2_2"/>
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="c10">p</font></span>)</span> <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/><b>then consider </b><font color="Maroon" title="c15">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E21:108_1_2_2"/><i><font color="Green" title="E74">A86</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">q</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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</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> <font color="Maroon" title="c10">p</font> implies for <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds not <font color="Maroon" title="c15">q</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> <font color="Maroon" title="c10">p</font></span>)</span> <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="Olive" title="b1">d</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon" title="c15">q</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="c8">z0</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> holds <br/>( <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c15">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="#E18:108_1_2_2"><i><font color="Green" title="E73">A85</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">u4</font> = <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">q</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E23:108_1_2_2"/><i><font color="Green" title="E75">A87</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">s</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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="c10">p</font> &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b2">s</font>,<font color="Maroon" title="c16">u4</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</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="c10">p</font> &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Maroon" title="c16">u4</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>let </b><font color="Maroon" title="c18">s</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</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="c10">p</font> &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s</font> implies <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c18">s</font>,<font color="Maroon" title="c16">u4</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<b>assume </b><a NAME="E1:108_1_2_2_2"/><i><font color="Green" title="E76">A88</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</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="c10">p</font> &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c18">s</font>,<font color="Maroon" title="c16">u4</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:108_1_2_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:108_1_2_2_2"/><b>then </b><i><font color="Green" title="E77">A89</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="c15">q</font>
 
<b>by </b><i><a class="txt" href="#E21:108_1_2_2"><i><font color="Green" title="E74">A86</font></i></a>, <a class="txt" href="#E1:108_1_2_2_2"><i><font color="Green" title="E76">A88</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">s2</font>, <font color="Olive" title="b2">u2</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; $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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s2</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b1">s2</font>,<font color="Olive" title="b2">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E4:108_1_2_2_2"/><i><font color="Green" title="E78">A90</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E5:108_1_2_2_2"/><i><font color="Green" title="E79">A91</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:108_1_2_2_2_1"/><i><font color="Green" title="E80">A92</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:108_1_2_2_2_1"/>
for <font color="Olive" title="b1">s2</font>, <font color="Olive" title="b2">u2</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b3">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k2</font> &amp; <font color="Olive" title="b3">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">k</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s2</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b1">s2</font>,<font color="Olive" title="b2">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">s2</font>, <font color="Maroon" title="c21">u2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c21">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k2</font> &amp; <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">k</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s2</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>let </b><font color="Maroon" title="c22">k2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k2</font> &amp; <font color="Maroon" title="c22">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">k</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s2</font> implies <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>





<b>assume </b><a NAME="E1:108_1_2_2_2_1_1"/><i><font color="Green" title="E81">A93</font></i>: 
( <font color="Maroon" title="c21">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k2</font> &amp; <font color="Maroon" title="c22">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">k</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:108_1_2_2_2_1_1"/><i><font color="Green" title="E82">A94</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:108_1_2_2_2_1_1"/><i><font color="Green" title="E83">A95</font></i>: 
<font color="Maroon" title="c22">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 
<b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_2_2_1_1_1"/>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_2_2_1_1_1_1"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:108_1_2_2_2_1_1_1_1"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_2_2_1_1_1_2"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:108_1_2_2_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E84">A96</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="c19">k</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="c23">s1</font> = <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c24">a</font> = <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font>, <font color="Maroon" title="c25">b</font> = <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E5:108_1_2_2_2_1_1_1_2"/><i><font color="Green" title="E85">A97</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c19">k</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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c25">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font> )
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="txt" href="#E2:108_1_2_2_2_1_1"><i><font color="Green" title="E82">A94</font></i></a>, <a class="txt" href="#E2:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E84">A96</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:108_1_2_2_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E86">A98</font></i>: 
( <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c15">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E21:108_1_2_2"><i><font color="Green" title="E74">A86</font></i></a></i>;<br/><a NAME="E7:108_1_2_2_2_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:108_1_2_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c21">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="txt" href="#E21:108_1_2_2"><i><font color="Green" title="E74">A86</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>then </b><i><font color="Green" title="E87">A99</font></i>: <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> = 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c24">a</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c25">b</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="#E6:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E86">A98</font></i></a>, <a class="ref" href="euclid_4.html#T31" title="EUCLID_4:th.31">EUCLID_4:31</a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c24">a</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c25">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T26" title="EUCLID_4:th.26">EUCLID_4:26</a></i>
;<br/><a NAME="E10:108_1_2_2_2_1_1_1_2"/><i><font color="Green" title="E88">A100</font></i>: 
<font color="Maroon" title="c19">k</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="c10">p</font>
 <b>by </b><i><a class="txt" href="#E5:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E85">A97</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_2_1_1_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_2_2_1_1_1_2_2"/>
( <font color="Maroon" title="c22">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font> or <font color="Maroon" title="c22">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">k</font> )
 </span><b>by </b><i><a class="txt" href="#E3:108_1_2_2_2_1_1"><i><font color="Green" title="E83">A95</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_2_1_1_1_2_2_1"><b>suppose </b></a><a NAME="E1:108_1_2_2_2_1_1_1_2_2_1"/><i><font color="Green" title="E89">A101</font></i>: 
<font color="Maroon" title="c22">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:108_1_2_2_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E90">A102</font></i>: 
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k2</font> &amp; <font color="Maroon" title="c22">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c19">k</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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s2</font> )
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="txt" href="#E2:108_1_2_2_2_1_1"><i><font color="Green" title="E82">A94</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:108_1_2_2_2_1_1_1_2_2_1"/><i><font color="Green" title="E91">A103</font></i>: 
<font color="Maroon" title="c22">k2</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="c10">p</font>
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E88">A100</font></i></a>, <a class="txt" href="#E1:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E89">A101</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:108_1_2_2_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E92">A104</font></i>: 
<font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k2</font>
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E5:108_1_2_2_2_1_1_1_2_2_1"/>
<font color="Maroon" title="c22">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="txt" href="#E3:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E91">A103</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:108_1_2_2_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E93">A105</font></i>: 
<font color="Maroon" title="c22">k2</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> <font color="Maroon" title="c11">p0</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:108_1_2_2_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E94">A106</font></i>: 
<font color="Maroon" title="c20">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p</font>
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="txt" href="#E4:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E92">A104</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:108_1_2_2_2_1_1_1_2_2_1"/><i><font color="Green" title="E95">A107</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="txt" href="#E1:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E89">A101</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:108_1_2_2_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E96">A108</font></i>: 
<font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E88">A100</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E10:108_1_2_2_2_1_1_1_2_2_1"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E95">A107</font></i></a>, <a class="txt" href="#E10:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E88">A100</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:108_1_2_2_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E97">A109</font></i>: 
<font color="Maroon" title="c19">k</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> <font color="Maroon" title="c11">p0</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:108_1_2_2_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E98">A110</font></i>: 
<font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="#E9:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E96">A108</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:108_1_2_2_2_1_1_1_2_2_1"/>
<font color="Maroon" title="c20">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">b</font>
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="txt" href="#E1:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E89">A101</font></i></a>, <a class="txt" href="#E4:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E92">A104</font></i></a>, <a class="txt" href="#E6:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E93">A105</font></i></a>, <a class="txt" href="#E9:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E96">A108</font></i></a>, <a class="txt" href="#E11:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E97">A109</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E14:108_1_2_2_2_1_1_1_2_2_1"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c25">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="#E7:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E94">A106</font></i></a>, <a class="txt" href="#E12:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E98">A110</font></i></a>, <a class="ref" href="euclid_7.html#D3" target="_self" title="EUCLID_7:def.3">Def4</a></i>;<br/><b>hence </b><a NAME="E15:108_1_2_2_2_1_1_1_2_2_1"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1"><i><font color="Green" title="E80">A92</font></i></a>, <a class="txt" href="#E9:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E87">A99</font></i></a>, <a class="txt" href="#E2:108_1_2_2_2_1_1_1_2_2_1"><i><font color="Green" title="E90">A102</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_2_1_1_1_2_2_2"><b>suppose </b></a><a NAME="E1:108_1_2_2_2_1_1_1_2_2_2"/><i><font color="Green" title="E89">A111</font></i>: 
<font color="Maroon" title="c22">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:108_1_2_2_2_1_1_1_2_2_2"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E84">A96</font></i></a>, <a class="txt" href="#E10:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E88">A100</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:108_1_2_2_2_1_1_1_2_2_2"/><b>then </b><i><font color="Green" title="E90">A112</font></i>: 
<font color="Maroon" title="c19">k</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> <font color="Maroon" title="c11">p0</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:108_1_2_2_2_1_1_1_2_2_2"/>
<font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E84">A96</font></i></a>, <a class="txt" href="#E10:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E88">A100</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E5:108_1_2_2_2_1_1_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">p0</font>
 <b>by </b><i><a class="txt" href="#E3:108_1_2_2_2_1_1_1_2_2_2"><i><font color="Green" title="E90">A112</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:108_1_2_2_2_1_1_1_2_2_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c25">b</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="euclid_7.html#D4" target="_self" title="EUCLID_7:def.4">Def5</a></i>;<br/><a NAME="E7:108_1_2_2_2_1_1_1_2_2_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c25">b</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><b>hence </b><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c21">u2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> = 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c25">b</font>,<font color="Maroon" title="c24">a</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:108_1_2_2_2_1_1"><i><font color="Green" title="E81">A93</font></i></a>, <a class="txt" href="#E9:108_1_2_2_2_1_1_1_2"><i><font color="Green" title="E87">A99</font></i></a>, <a class="txt" href="#E1:108_1_2_2_2_1_1_1_2_2_2"><i><font color="Green" title="E89">A111</font></i></a>, <a class="ref" href="euclid_2.html#T12" title="EUCLID_2:th.12">EUCLID_2:12</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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>
<b>hence </b><a NAME="E3:108_1_2_2_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:108_1_2_2_2"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:108_1_2_2_2"><i><font color="Green" title="E78">A90</font></i></a>, <a class="txt" href="#E5:108_1_2_2_2"><i><font color="Green" title="E79">A91</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:108_1_2_2_2"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c18">s</font>,<font color="Maroon" title="c16">u4</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E21:108_1_2_2"><i><font color="Green" title="E74">A86</font></i></a>, <a class="txt" href="#E1:108_1_2_2_2"><i><font color="Green" title="E76">A88</font></i></a>, <a class="txt" href="#E3:108_1_2_2_2"><i><font color="Green" title="E77">A89</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c17">B2</font> = <font color="Maroon" title="c4">B</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E5:108_1_2"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="#E10:108_1_2"><i><font color="Green" title="E64">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">a2</font> = <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c10">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>, <font color="Maroon" title="c19">b2</font> = <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c10">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><b>set </b><font color="Maroon" title="c20">aq</font> = <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c21">bq</font> = <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span>;<br/><a NAME="E26:108_1_2_2"/><i><font color="Green" title="E76">A113</font></i>: 
( <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font></span>)</span> )
 ;<br/><a NAME="E27:108_1_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</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="c10">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E28:108_1_2_2"/><b>then </b><i><font color="Green" title="E77">A114</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E12:108_1_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="#E21:108_1_2_2"><i><font color="Green" title="E74">A86</font></i></a>, <a class="txt" href="#E26:108_1_2_2"><i><font color="Green" title="E76">A113</font></i></a>, <a class="ref" href="euclid.html#T11" title="EUCLID:th.11">EUCLID:11</a></i>;<br/><a NAME="E29:108_1_2_2"/><i><font color="Green" title="E78">A115</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>set </b><font color="Maroon" title="c22">u0</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font>;<br/><i><font color="Green" title="E79">A116</font></i>: <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14" title="EUCLID:th.14">EUCLID:14</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="#E28:108_1_2_2"><i><font color="Green" title="E77">A114</font></i></a>, <a class="txt" href="#E29:108_1_2_2"><i><font color="Green" title="E78">A115</font></i></a>, <a class="ref" href="xcmplx_1.html#T107" title="XCMPLX_1:th.107">XCMPLX_1:107</a></i>
;<br/><a NAME="E31:108_1_2_2"/><i><font color="Green" title="E80">A117</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">s</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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="c10">p</font> &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b2">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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="c10">p</font> &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>let </b><font color="Maroon" title="c24">s</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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="c10">p</font> &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s</font> implies <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<b>assume </b><a NAME="E1:108_1_2_2_4"/><i><font color="Green" title="E81">A118</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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="c10">p</font> &amp; <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:108_1_2_2_4"/><i><font color="Green" title="E82">A119</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">s</font>,<font color="Maroon" title="c16">u4</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
 
<b>by </b><i><a class="ref" href="euclid_4.html#T27" title="EUCLID_4:th.27">EUCLID_4:27</a></i>;<br/>
<a NAME="E3:108_1_2_2_4"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">s</font>,<font color="Maroon" title="c16">u4</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:108_1_2_2_4"><i><font color="Green" title="E81">A118</font></i></a>, <a class="txt" href="#E23:108_1_2_2"><i><font color="Green" title="E75">A87</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:108_1_2_2_4"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E2:108_1_2_2_4"><i><font color="Green" title="E82">A119</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c23">B3</font> = <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E33:108_1_2_2"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B3</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B3</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">x</font>, <font color="Maroon" title="c25">y</font> be  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B3</font> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B3</font> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">y</font> implies <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:108_1_2_2_5"/><i><font color="Green" title="E81">A120</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B3</font> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B3</font> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_2_5_1"/>
( ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) or ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) or ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) or ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="#E1:108_1_2_2_5"><i><font color="Green" title="E81">A120</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_5_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_2_5_1_1"/>
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:108_1_2_2_5_1_1"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_5"><i><font color="Green" title="E81">A120</font></i></a>, <a class="ref" href="euclid_7.html#D3" target="_self" title="EUCLID_7:def.3">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_5_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_2_5_1_2"/><i><font color="Green" title="E82">A121</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:108_1_2_2_5_1_2"/><b>then </b><i><font color="Green" title="E83">A122</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:108_1_2_2_5_1_2"/><i><font color="Green" title="E84">A123</font></i>: 
( <font color="Maroon" title="c26">x3</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> <font color="Maroon" title="c11">p0</font> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x3</font> )
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="#E1:108_1_2_2_5_1_2"><i><font color="Green" title="E82">A121</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:108_1_2_2_5_1_2"/><i><font color="Green" title="E85">A124</font></i>: 
<font color="Maroon" title="c26">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:108_1_2_2_5_1_2"><i><font color="Green" title="E84">A123</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">j</font> = <font color="Maroon" title="c26">x3</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E4:108_1_2_2_5_1_2"><i><font color="Green" title="E84">A123</font></i></a></i>;<br/><a NAME="E7:108_1_2_2_5_1_2"/><i><font color="Green" title="E86">A125</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">j</font> &amp; <font color="Maroon" title="c27">j</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">p0</font> )
 <b>by </b><i><a class="txt" href="#E5:108_1_2_2_5_1_2"><i><font color="Green" title="E85">A124</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:108_1_2_2_5_1_2"/><b>then </b>
<font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>hence </b><a NAME="E9:108_1_2_2_5_1_2"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E31:108_1_2_2"><i><font color="Green" title="E80">A117</font></i></a>, <a class="txt" href="#E2:108_1_2_2_5_1_2"><i><font color="Green" title="E83">A122</font></i></a>, <a class="txt" href="#E4:108_1_2_2_5_1_2"><i><font color="Green" title="E84">A123</font></i></a>, <a class="txt" href="#E7:108_1_2_2_5_1_2"><i><font color="Green" title="E86">A125</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_5_1_3"><b>suppose </b></a><a NAME="E1:108_1_2_2_5_1_3"/><i><font color="Green" title="E82">A126</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:108_1_2_2_5_1_3"/><b>then </b><i><font color="Green" title="E83">A127</font></i>: 
( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">y3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:108_1_2_2_5_1_3"/><i><font color="Green" title="E84">A128</font></i>: 
( <font color="Maroon" title="c26">y3</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> <font color="Maroon" title="c11">p0</font> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y3</font> )
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="#E1:108_1_2_2_5_1_3"><i><font color="Green" title="E82">A126</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:108_1_2_2_5_1_3"/><i><font color="Green" title="E85">A129</font></i>: 
<font color="Maroon" title="c26">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:108_1_2_2_5_1_3"><i><font color="Green" title="E84">A128</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">j</font> = <font color="Maroon" title="c26">y3</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E4:108_1_2_2_5_1_3"><i><font color="Green" title="E84">A128</font></i></a></i>;<br/><a NAME="E7:108_1_2_2_5_1_3"/><i><font color="Green" title="E86">A130</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">j</font> &amp; <font color="Maroon" title="c27">j</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">p0</font> )
 <b>by </b><i><a class="txt" href="#E5:108_1_2_2_5_1_3"><i><font color="Green" title="E85">A129</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:108_1_2_2_5_1_3"/><b>then </b>
<font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>hence </b><a NAME="E9:108_1_2_2_5_1_3"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E31:108_1_2_2"><i><font color="Green" title="E80">A117</font></i></a>, <a class="txt" href="#E2:108_1_2_2_5_1_3"><i><font color="Green" title="E83">A127</font></i></a>, <a class="txt" href="#E4:108_1_2_2_5_1_3"><i><font color="Green" title="E84">A128</font></i></a>, <a class="txt" href="#E7:108_1_2_2_5_1_3"><i><font color="Green" title="E86">A130</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_5_1_4"><b>suppose </b></a><a NAME="E1:108_1_2_2_5_1_4"/>
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:108_1_2_2_5_1_4"/><b>then </b>
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:108_1_2_2_5_1_4"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:108_1_2_2_5"><i><font color="Green" title="E81">A120</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E34:108_1_2_2"/><b>then </b><i><font color="Green" title="E81">A131</font></i>: 
<font color="Maroon" title="c23">B3</font> is <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a>
 <b>by </b><i><a class="ref" href="euclid_7.html#D3" target="_self" title="EUCLID_7:def.3">Def4</a></i>;<br/><a NAME="E35:108_1_2_2"/>
<font color="Maroon" title="c23">B3</font> is <a href="euclid_7.html#V2" title="EUCLID_7:attr.2">R-normal</a>
 <b>by </b><i><a class="txt" href="#E30:108_1_2_2"><i><font color="Green" title="E79">A116</font></i></a>, <a class="ref" href="euclid_7.html#T18" target="_self" title="EUCLID_7:th.18">Th20</a></i>;<br/><a NAME="E36:108_1_2_2"/><b>then </b><i><font color="Green" title="E82">A132</font></i>: 
<font color="Maroon" title="c5">D0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">B3</font>
 <b>by </b><i><a class="txt" href="#E3:108_1_2"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="#E34:108_1_2_2"><i><font color="Green" title="E81">A131</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E37:108_1_2_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E38:108_1_2_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E36:108_1_2_2"><i><font color="Green" title="E82">A132</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E40:108_1_2_2"/><i><font color="Green" title="E83">A133</font></i>: 
( <font color="Maroon" title="c24">x3</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> <font color="Maroon" title="c11">p0</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x3</font> )
 <b>by </b><i><a class="txt" href="#E10:108_1_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E41:108_1_2_2"/><i><font color="Green" title="E84">A134</font></i>: 
<font color="Maroon" title="c24">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E40:108_1_2_2"><i><font color="Green" title="E83">A133</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">j</font> = <font color="Maroon" title="c24">x3</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E40:108_1_2_2"><i><font color="Green" title="E83">A133</font></i></a></i>;<br/><a NAME="E43:108_1_2_2"/><i><font color="Green" title="E85">A135</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font> &amp; <font color="Maroon" title="c25">j</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">p0</font> )
 <b>by </b><i><a class="txt" href="#E41:108_1_2_2"><i><font color="Green" title="E84">A134</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E44:108_1_2_2"/><b>then </b>
<font color="Maroon" title="c11">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E45:108_1_2_2"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c16">u4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">u4</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E31:108_1_2_2"><i><font color="Green" title="E80">A117</font></i></a>, <a class="txt" href="#E40:108_1_2_2"><i><font color="Green" title="E83">A133</font></i></a>, <a class="txt" href="#E43:108_1_2_2"><i><font color="Green" title="E85">A135</font></i></a></i>;<br/><b>hence </b><a NAME="E46:108_1_2_2"/>
<font color="Maroon" title="c4">B</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="#E30:108_1_2_2"><i><font color="Green" title="E79">A116</font></i></a>, <a class="ref" href="euclid_4.html#T21" title="EUCLID_4:th.21">EUCLID_4:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E13:108_1_2"/>
<font color="Maroon" title="c2">D</font> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="euclid_7.html#K9" title="EUCLID_7:func.9">REAL-L</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="#E11:108_1_2"><i><font color="Green" title="E65">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
