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<span class="kw">consider </span><font color="Maroon" title="c1">t</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon">F<sub>4</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:59_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
( <font color="Maroon" title="c1">t</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">F<sub>2</sub></font>() &amp; <font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E37">W2</font></span></a>, <a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">t</font> = <font color="Maroon" title="c1">t</font> as  <font color="Maroon">F<sub>1</sub></font>() <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon">F<sub>4</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="stacks_1.html#T23" target="_self" title="STACKS_1:th.23">Lem2</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $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="c2">t</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1] );<br/>
<a NAME="E4:59_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T24" title="FINSEQ_3:th.24">FINSEQ_3:24</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E41">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:59_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:59_1_1_1"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
( <font color="Maroon" title="c3">i</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="c2">t</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:59_1_1_1"/>
( <font color="Maroon" title="c3">i</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="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:59_1_1_1"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E36">W1</font></span></a>, <a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:59_1_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<a NAME="E5:59_1_1_1"/>
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E6:59_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A9</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">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="c2">t</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1_1_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:59_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A7</font></span>: 
( <font color="Maroon" title="c3">i</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="c2">t</font> &amp; 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="c2">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a>, <a class="ref" href="finseq_3.html#T32" title="FINSEQ_3:th.32">FINSEQ_3:32</a></span>;<br/>
<a NAME="E8:59_1_1_1"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
( <font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c2">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E7:59_1_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E9:59_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E7:59_1_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:59_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E38">W3</font></span></a>, <a class="txt" href="#E1:59_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E1:59_1_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E7:59_1_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E6:59_1_1_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E8:59_1_1_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="rewrite1.html#T17" title="REWRITE1:th.17">REWRITE1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E6:59_1"/><span class="lab"><font color="Green" title="E42">BB</font></span>: 
 for <font color="Olive" title="b1">i</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">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E5:59_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>);<br/></span>
<a NAME="E7:59_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">t</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="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:59_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>3</sub></font>()]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E6:59_1"><span class="lab"><font color="Green" title="E42">BB</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
