<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">m</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/>  for <font color="Olive" title="b3">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><br/>  for <font color="Olive" title="b4">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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b4">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">z</font> holds <br/>( <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Olive" title="b3">z</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b4">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b2">y</font></span>)</span>) &amp; <font color="Olive" title="b3">z</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Olive" title="b3">z</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b4">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">x</font></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><br/>  for <font color="Olive" title="b2">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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">z</font> holds <br/>( <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b2">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>) &amp; <font color="Olive" title="b1">z</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b2">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">x</font></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">z</font> holds <br/>( <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>) &amp; <font color="Maroon" title="c4">z</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">x</font></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">z</font> implies ( <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>) &amp; <font color="Maroon" title="c4">z</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">x</font></span>)</span>) ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> )
 <span class="kw">and </span><br/><a NAME="E2:24"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>) &amp; <font color="Maroon" title="c4">z</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">x</font></span>)</span>) )</span><br/>

<a NAME="E3:24"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="pdiff_7.html#E16"><span class="lab"><font color="Green" title="E15">Lm6</font></span></a></span>;<br/>
<a NAME="E4:24"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T30" title="FUNCT_7:th.30">FUNCT_7:30</a></span>;<br/>
<a NAME="E5:24"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">z</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E6:24"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E7:24"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E8:24"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="finseq_1.html#T89" title="FINSEQ_1:th.89">FINSEQ_1:89</a></span>;<br/>
<a NAME="E9:24"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_7.html#T5" title="FINSEQ_7:th.5">FINSEQ_7:5</a></span>;<br/>
<a NAME="E11:24"/>
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> implies <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c6">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:24_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:24_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>;<br/>
<a NAME="E5:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E6:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A11</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span>
 
;<br/>
<a NAME="E7:24_1"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">z</font> implies <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E2:24_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">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="c4">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:24_1"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E9:24_1"/>
( <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> &amp; <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E10:24_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><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="E1:24_1_2"/>
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> or  not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_2_1"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="pdiff_7.html#T9" target="_self" title="PDIFF_7:th.9">Th9</a></span>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E11:24_1"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="txt" href="#E1:24_1_2_1"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="pdiff_7.html#E17"><span class="lab"><font color="Green" title="E16">Lm7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E1:24_1_2_1"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="pdiff_7.html#E17"><span class="lab"><font color="Green" title="E16">Lm7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E22">A2</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:24_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E1:24_1_2_1"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_2_2"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
 not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span><br/><div class="add"><a NAME="E2:24_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="pdiff_7.html#E17"><span class="lab"><font color="Green" title="E16">Lm7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_1_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="txt" href="#E1:24_1_2_2"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:24"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">z</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">x</font></span>)</span>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> =  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E14:24"/>
<font color="Maroon" title="c4">z</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="rvsum_1.html#T35" title="RVSUM_1:th.35">RVSUM_1:35</a></span>;<br/>
<a NAME="E15:24"/><span class="kw">then </span>
<font color="Maroon" title="c4">z</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="pdiff_7.html#T10" target="_self" title="PDIFF_7:th.10">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:24"/>
<font color="Maroon" title="c4">z</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">x</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
