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

<b>let </b><font color="Maroon" title="c3">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<a NAME="E1:34_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F1</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c2">F2</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> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">F3</font> = <font color="Maroon" title="c1">F1</font>, <font color="Maroon" title="c5">F4</font> = <font color="Maroon" title="c2">F2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="rvsum_1.html#E31"><i><font color="Green" title="E10">Lm4</font></i></a></i>;<br/>
<a NAME="E3:34_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:34_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">F3</font></span>)</span>,<span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">F4</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E5:34_1_1"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="funcop_1.html#K3" title="FUNCOP_1:func.3">.:</a> <font color="Maroon" title="c1">F1</font>,<font color="Maroon" title="c2">F2</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> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T84" title="FUNCOP_1:th.84">FUNCOP_1:84</a></i>;<br/>
<b>thus </b><a NAME="E6:34_1_1"/>
( <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F1</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c2">F2</font> implies <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="funcop_1.html#K3" title="FUNCOP_1:func.3">.:</a> <font color="Maroon" title="c1">F1</font>,<font color="Maroon" title="c2">F2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1_1"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F1</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c2">F2</font>
 ;<br/>

<a NAME="E2:34_1_1_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="funcop_1.html#K3" title="FUNCOP_1:func.3">.:</a> <font color="Maroon" title="c1">F1</font>,<font color="Maroon" title="c2">F2</font></span>)</span> holds <br/><font color="Maroon" title="c3">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="funcop_1.html#K3" title="FUNCOP_1:func.3">.:</a> <font color="Maroon" title="c1">F1</font>,<font color="Maroon" title="c2">F2</font></span>)</span>
 ;<br/>

<b>hence </b><font color="Maroon" title="c3">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">z</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="c2">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E1:34_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="rvsum_1.html#E5:34_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="rvsum_1.html#E1:34_1_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<br/>.= 
<a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D10" title="BINOP_2:def.10">BINOP_2:def 10</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="funcop_1.html#K3" title="FUNCOP_1:func.3">.:</a> <font color="Maroon" title="c1">F1</font>,<font color="Maroon" title="c2">F2</font>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E1:34_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="rvsum_1.html#E5:34_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="rvsum_1.html#E1:34_1_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="funcop_1.html#T27" title="FUNCOP_1:th.27">FUNCOP_1:27</a></i>;<br/>


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

<b>assume </b><a NAME="E7:34_1_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="funcop_1.html#K3" title="FUNCOP_1:func.3">.:</a> <font color="Maroon" title="c1">F1</font>,<font color="Maroon" title="c2">F2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:34_1_1_2"/>
<font color="Maroon" title="c6">c</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="c3">F</font>
 ;<br/><b>hence </b><font color="Maroon" title="c3">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font> = 
<a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E7:34_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</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="c2">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D10" title="BINOP_2:def.10">BINOP_2:def 10</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:34_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F1</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c2">F2</font>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E1:34_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="rvsum_1.html#E5:34_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="rvsum_1.html#E7:34_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="valued_1.html#T14" title="VALUED_1:th.14">VALUED_1:14</a></i>;<br/>


</div>
