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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">h1</font>, <font color="Maroon" title="c4">h2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>







<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c3">h1</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">h2</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">r1</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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_2.html#D4" target="_self" title="RFUNCT_2:def.4">RFUNCT_2:def 4</a><br/><b>let </b><font color="Maroon" title="c6">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> ;<br/>



<b>assume </b><a NAME="E2:70"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 ;<br/>

<a NAME="E3:70"/><b>then </b><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:70"/><b>then </b><i><font color="Green" title="E33">A4</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:70"/>
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E3:70"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>
<a NAME="E6:70"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:70"/><b>then </b>
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E4:70"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:70"/><b>then </b><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:70"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="rfunct_2.html#E2:70"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="rfunct_2.html#D4" target="_self" title="RFUNCT_2:def.4">Def4</a></i>;<br/>
<a NAME="E9:70"/>
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E4:70"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="rfunct_2.html#E6:70"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:70"/><b>then </b>
<font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:70"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="partfun2.html#T77" title="PARTFUN2:th.77">PARTFUN2:77</a></i>;<br/>
<a NAME="E11:70"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E8:70"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E12:70"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E3:70"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:70"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font>
 <b>by </b><i><a class="txt" href="rfunct_2.html#E3:70"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>


</div>
