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<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">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">C</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>









<b>thus </b><a NAME="E1:99"/>
( <font color="Maroon" title="c4">f1</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c4">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">f2</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_1"/><i><font color="Green" title="E59">A1</font></i>: 
( <font color="Maroon" title="c4">f1</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<a NAME="E2:99_1"/><b>then </b>
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c5">f2</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T107" target="_self" title="RFUNCT_1:th.107">Th107</a></i>;<br/>
<a NAME="E3:99_1"/><b>then </b>
<font color="Maroon" title="c4">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</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>
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:99_1"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T110" target="_self" title="RFUNCT_1:th.110">Th110</a></i>;<br/>
<b>hence </b><a NAME="E4:99_1"/>
<font color="Maroon" title="c4">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">f2</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</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>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T40" target="_self" title="RFUNCT_1:th.40">Th40</a></i>;<br/>


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

<b>thus </b><a NAME="E2:99"/>
( <font color="Maroon" title="c4">f1</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c4">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">f2</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_2"/><i><font color="Green" title="E59">A2</font></i>: 
( <font color="Maroon" title="c4">f1</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<a NAME="E2:99_2"/><b>then </b>
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c5">f2</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T107" target="_self" title="RFUNCT_1:th.107">Th107</a></i>;<br/>
<a NAME="E3:99_2"/><b>then </b>
<font color="Maroon" title="c4">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</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>
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:99_2"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="rfunct_1.html#T110" target="_self" title="RFUNCT_1:th.110">Th110</a></i>;<br/>
<b>hence </b><a NAME="E4:99_2"/>
<font color="Maroon" title="c4">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">f2</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</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>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T40" target="_self" title="RFUNCT_1:th.40">Th40</a></i>;<br/>


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

<b>assume </b><a NAME="E3:99"/><i><font color="Green" title="E59">A3</font></i>: 
( <font color="Maroon" title="c4">f1</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<a NAME="E4:99"/><b>then </b>
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c5">f2</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T107" target="_self" title="RFUNCT_1:th.107">Th107</a></i>;<br/>
<a NAME="E5:99"/><b>then </b>
<font color="Maroon" title="c4">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</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>
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E3:99"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="rfunct_1.html#T110" target="_self" title="RFUNCT_1:th.110">Th110</a></i>;<br/>
<b>hence </b><a NAME="E6:99"/>
<font color="Maroon" title="c4">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">f2</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</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>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T40" target="_self" title="RFUNCT_1:th.40">Th40</a></i>;<br/>

<a NAME="E7:99"/><i><font color="Green" title="E60">A4</font></i>: 
<font color="Maroon" title="c5">f2</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E3:99"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="rfunct_1.html#T108" target="_self" title="RFUNCT_1:th.108">Th108</a></i>;<br/>
<b>hence </b><a NAME="E8:99"/>
<font color="Maroon" title="c5">f2</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f1</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</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>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E3:99"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="rfunct_1.html#T101" target="_self" title="RFUNCT_1:th.101">Th101</a></i>;<br/>

<b>thus </b><a NAME="E9:99"/>
<font color="Maroon" title="c4">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c5">f2</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</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>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E3:99"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="rfunct_1.html#E7:99"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="rfunct_1.html#T101" target="_self" title="RFUNCT_1:th.101">Th101</a></i>;<br/>


</div>
