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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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"> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">R</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">r1</font>, <font color="Maroon" title="c3">r2</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"> ( <font color="Maroon" title="c2">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r2</font> implies <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57_1"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c2">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:57_1"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c3">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:57_1"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r2</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="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r2</font></span><br/><a NAME="E4:57_1"/>
<font color="Maroon" title="c2">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E22">A1</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="E5:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>;<br/><a NAME="E6:57_1"/>
<font color="Maroon" title="c3">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:57_1"/>
<span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</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>
<span class="kw">hence </span><a NAME="E2:57"/>
<span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">R</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_2.html#T20" target="_self" title="RFUNCT_2:th.20">Th43</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>
