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<span class="kw">let </span><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">Q</font> be   <a href="rewrite1.html#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a>  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R11" title="REWRITE1:pred.11">commutes_with</a> <font color="Maroon" title="c2">Q</font> implies <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> is  <a href="rewrite1.html#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R11" title="REWRITE1:pred.11">commutes_with</a> <font color="Maroon" title="c2">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> is  <a href="rewrite1.html#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a> </span><br/>

<a NAME="E2:84"/>
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> is  <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite1.html#D22" target="_self" title="REWRITE1:def.22">REWRITE1:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R6" title="REWRITE1:pred.6">are_divergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> implies <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c5">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E2:84_1"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E3:84_1"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite1.html#D8" target="_self" title="REWRITE1:def.8">REWRITE1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font><span class="kw"> such that </span><br/><a NAME="E5:84_1"/><span class="lab"><font color="Green" title="E69">A4</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E6:84_1"/><span class="lab"><font color="Green" title="E70">A5</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</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="c6">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="ref" href="rewrite1.html#D3" target="_self" title="REWRITE1:def.3">Def3</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <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="c6">p</font> implies <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">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="c6">p</font> implies <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> ) &amp; <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font> implies <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:84_1_1"/><span class="lab"><font color="Green" title="E71">A6</font></span>: 
( <font color="Maroon" title="c7">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="c6">p</font> implies <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> )
 <span class="kw">and </span><br/><a NAME="E2:84_1_1"/><span class="lab"><font color="Green" title="E72">A7</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:84_1_1_1"/>
( <font color="Maroon" title="c7">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="c7">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:84_1_1_1_1"/>
<font color="Maroon" title="c7">i</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:84_1_1_1_1"/>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="txt" href="#E5:84_1"><span class="lab"><font color="Green" title="E69">A4</font></span></a>, <a class="ref" href="rewrite1.html#T37" target="_self" title="REWRITE1:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:84_1_1_1_2"/><span class="lab"><font color="Green" title="E73">A8</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font></span><br/><div class="add"><a NAME="E2:84_1_1_1_2"/><span class="lab"><font color="Green" title="E74">A9</font></span>: 
<font color="Maroon" title="c7">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="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_1"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="txt" href="rewrite1.html#E3"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:84_1_1_1_2"/><span class="lab"><font color="Green" title="E75">A10</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">d</font>
 <span class="kw">and </span><br/><a NAME="E5:84_1_1_1_2"/><span class="lab"><font color="Green" title="E76">A11</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c8">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1_1"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="txt" href="#E1:84_1_1_1_2"><span class="lab"><font color="Green" title="E73">A8</font></span></a>, <a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a>, <a class="txt" href="rewrite1.html#E4"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">q</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font><span class="kw"> such that </span><br/><a NAME="E7:84_1_1_1_2"/><span class="lab"><font color="Green" title="E77">A12</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E8:84_1_1_1_2"/><span class="lab"><font color="Green" title="E78">A13</font></span>: 
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</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="c9">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1_1_1_2"><span class="lab"><font color="Green" title="E75">A10</font></span></a>, <a class="ref" href="rewrite1.html#D3" target="_self" title="REWRITE1:def.3">Def3</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <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="c9">q</font> implies  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">e</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,<font color="Olive" title="b1">e</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,<font color="Olive" title="b1">e</font> ) ) );<br/><a NAME="E9:84_1_1_1_2"/>
<font color="Maroon" title="c7">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="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1_1_1_2"><span class="lab"><font color="Green" title="E73">A8</font></span></a>, <a class="txt" href="#E2:84_1_1_1_2"><span class="lab"><font color="Green" title="E74">A9</font></span></a>, <a class="txt" href="rewrite1.html#E4"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><a NAME="E10:84_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_1"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="ref" href="rewrite1.html#D2" target="_self" title="REWRITE1:def.2">Def2</a></span>;<br/><a NAME="E11:84_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A14</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c10">j</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="c9">q</font> implies  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">e</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Olive" title="b1">e</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Olive" title="b1">e</font> ) ) ) &amp; <font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font> implies  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">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">e</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">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">e</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:84_1_1_1_2_1"/><span class="lab"><font color="Green" title="E80">A15</font></span>: 
( <font color="Maroon" title="c10">j</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="c9">q</font> implies  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">e</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Olive" title="b1">e</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Olive" title="b1">e</font> ) ) )
 <span class="kw">and </span><br/><a NAME="E2:84_1_1_1_2_1"/><span class="lab"><font color="Green" title="E81">A16</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">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">e</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">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">e</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> ) )</span><br/><a NAME="E3:84_1_1_1_2_1"/><span class="lab"><font color="Green" title="E82">A17</font></span>: 
<font color="Maroon" title="c10">j</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="c9">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A16</font></span></a>, <a class="txt" href="rewrite1.html#E3"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:84_1_1_1_2_1_1"/>
( <font color="Maroon" title="c10">j</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="c10">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:84_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E83">A18</font></span>: 
<font color="Maroon" title="c10">j</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">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">e</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">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">e</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> ) )</span><br/><div class="add"><a NAME="E2:84_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E84">A19</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T13" target="_self" title="REWRITE1:th.13">Th13</a></span>;<br/><a NAME="E3:84_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:84_1_1_1_2"><span class="lab"><font color="Green" title="E77">A12</font></span></a>, <a class="txt" href="#E11:84_1_1_1_2"><span class="lab"><font color="Green" title="E79">A14</font></span></a>, <a class="txt" href="#E1:84_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A18</font></span></a>, <a class="ref" href="rewrite1.html#T16" target="_self" title="REWRITE1:th.16">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E4:84_1_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">e</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">e</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">e</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:84_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E83">A20</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">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">e</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">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">e</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:84_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E84">A21</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">e</font>
 <span class="kw">and </span><br/><a NAME="E4:84_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E85">A22</font></span>: 
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">e</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1_1_1_2_1"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="txt" href="#E3:84_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A17</font></span></a>, <a class="txt" href="rewrite1.html#E4"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E86">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:84_1_1_1_2_1_1_2_1"/>
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:84_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E87">A24</font></span>: 
<font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">x</font>
 <span class="kw">and </span><br/><a NAME="E4:84_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E88">A25</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="rewrite1.html#D18" target="_self" title="REWRITE1:def.18">Def18</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> ) )</span><br/><a NAME="E5:84_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E88">A25</font></span></a>, <a class="ref" href="rewrite1.html#T23" target="_self" title="REWRITE1:th.23">Th23</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:84_1_1_1_2_1_1_2_1"/>
( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A21</font></span></a>, <a class="txt" href="#E3:84_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E87">A24</font></span></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E87">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1_2_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:84_1_1_1_2_1_1_2_2"/>
( <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> ) )</span><br/><a NAME="E2:84_1_1_1_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="rewrite1.html#R6" title="REWRITE1:pred.6">are_divergent_wrt</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D8" target="_self" title="REWRITE1:def.8">Def8</a></span>;<br/><a NAME="E3:84_1_1_1_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D22" target="_self" title="REWRITE1:def.22">Def22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:84_1_1_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E88">A27</font></span>: 
<font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c12">x</font>
 <span class="kw">and </span><br/><a NAME="E6:84_1_1_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E89">A28</font></span>: 
<font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> ) )</span><br/><a NAME="E7:84_1_1_1_2_1_1_2_2"/>
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_1_1_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E88">A27</font></span></a>, <a class="ref" href="rewrite1.html#T23" target="_self" title="REWRITE1:th.23">Th23</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:84_1_1_1_2_1_1_2_2"/>
( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A21</font></span></a>, <a class="txt" href="#E6:84_1_1_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E89">A28</font></span></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E88">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1_2_1_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:84_1_1_1_2_1_1_2_3"/>
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> ) )</span><br/><a NAME="E2:84_1_1_1_2_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="rewrite1.html#R6" title="REWRITE1:pred.6">are_divergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D8" target="_self" title="REWRITE1:def.8">Def8</a></span>;<br/><a NAME="E3:84_1_1_1_2_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D22" target="_self" title="REWRITE1:def.22">Def22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:84_1_1_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E89">A30</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c12">x</font>
 <span class="kw">and </span><br/><a NAME="E6:84_1_1_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E90">A31</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> ) )</span><br/><a NAME="E7:84_1_1_1_2_1_1_2_3"/>
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_1_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E89">A30</font></span></a>, <a class="ref" href="rewrite1.html#T23" target="_self" title="REWRITE1:th.23">Th23</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:84_1_1_1_2_1_1_2_3"/>
( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A21</font></span></a>, <a class="txt" href="#E6:84_1_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E90">A31</font></span></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E89">A32</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1_2_1_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:84_1_1_1_2_1_1_2_4"/>
( <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:84_1_1_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E90">A33</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">x</font>
 <span class="kw">and </span><br/><a NAME="E4:84_1_1_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E91">A34</font></span>: 
<font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="rewrite1.html#D18" target="_self" title="REWRITE1:def.18">Def18</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> ) )</span><br/><a NAME="E5:84_1_1_1_2_1_1_2_4"/>
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1_1_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E91">A34</font></span></a>, <a class="ref" href="rewrite1.html#T23" target="_self" title="REWRITE1:th.23">Th23</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:84_1_1_1_2_1_1_2_4"/>
( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A21</font></span></a>, <a class="txt" href="#E3:84_1_1_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E90">A33</font></span></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:84_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">j</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="c9">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A17</font></span></a>, <a class="txt" href="#E1:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="txt" href="rewrite1.html#E4"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><a NAME="E10:84_1_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A16</font></span></a>, <a class="ref" href="rewrite1.html#D2" target="_self" title="REWRITE1:def.2">Def2</a></span>;<br/><a NAME="E11:84_1_1_1_2_1_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E12:84_1_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">e</font> &amp; ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">e</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">e</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A22</font></span></a>, <a class="txt" href="#E7:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A29</font></span></a>, <a class="txt" href="#E6:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E87">A26</font></span></a>, <a class="txt" href="#E5:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E86">A23</font></span></a>, <a class="txt" href="#E8:84_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A32</font></span></a>, <a class="ref" href="rewrite1.html#T16" target="_self" title="REWRITE1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E13:84_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A35</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>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#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/><a NAME="E14:84_1_1_1_2"/><span class="lab"><font color="Green" title="E81">A36</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="rewrite1.html#E2"><span class="lab"><font color="Green" title="E2">Lm1</font></span></a></span>;<br/><a NAME="E15:84_1_1_1_2"/><span class="lab"><font color="Green" title="E82">A37</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E14:84_1_1_1_2"><span class="lab"><font color="Green" title="E81">A36</font></span></a>, <a class="txt" href="#E13:84_1_1_1_2"><span class="lab"><font color="Green" title="E80">A35</font></span></a>);<br/></span><span class="kw">thus </span><a NAME="E16:84_1_1_1_2"/>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:84_1_1_1_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</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="c9">q</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">then </span><span class="kw">consider </span><font color="Maroon" title="c10">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:84_1_1_1_2_2"/><span class="lab"><font color="Green" title="E83">A38</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">e</font>
 <span class="kw">and </span><br/><a NAME="E4:84_1_1_1_2_2"/><span class="lab"><font color="Green" title="E84">A39</font></span>: 
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">e</font> or <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:84_1_1_1_2"><span class="lab"><font color="Green" title="E78">A13</font></span></a>, <a class="txt" href="#E15:84_1_1_1_2"><span class="lab"><font color="Green" title="E82">A37</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">e</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">REWRITE1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">e</font> &amp; <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c10">e</font> )</span><br/>

<a NAME="E5:84_1_1_1_2_2"/>
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1_1_1_2_2"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="ref" href="rewrite1.html#T23" target="_self" title="REWRITE1:th.23">Th23</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:84_1_1_1_2_2"/>
( <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">e</font> &amp; <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c10">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_1_1_1_2"><span class="lab"><font color="Green" title="E76">A11</font></span></a>, <a class="txt" href="#E3:84_1_1_1_2_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E8:84_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A40</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
;<br/>
<a NAME="E9:84_1"/><span class="lab"><font color="Green" title="E72">A41</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</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="c6">p</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/>
<a NAME="E10:84_1"/><span class="lab"><font color="Green" title="E73">A42</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="txt" href="rewrite1.html#E2"><span class="lab"><font color="Green" title="E2">Lm1</font></span></a></span>;<br/>
<a NAME="E11:84_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E10:84_1"><span class="lab"><font color="Green" title="E73">A42</font></span></a>, <a class="txt" href="#E8:84_1"><span class="lab"><font color="Green" title="E71">A40</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E12:84_1"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84_1"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="txt" href="#E9:84_1"><span class="lab"><font color="Green" title="E72">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:84"/>
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Q</font> is  <a href="rewrite1.html#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
