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<b>let </b><font color="Maroon" title="c1">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be  <a href="seq_1.html#V1" title="SEQ_1:attr.1">real-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>





<b>assume </b><a NAME="E1:82"/><i><font color="Green" title="E47">A1</font></i>: 
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">p1</font> = 0 as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p1</font>;<br/><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="E1:82_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seq_1.html#K12" title="SEQ_1:func.12">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:82_1"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seq_1.html#K12" title="SEQ_1:func.12">(#)</a> <font color="Maroon" title="c3">f</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="E3:82_1"/>
<font color="Maroon" title="c2">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="rfunct_1.html#E1:82"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:82_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seq_1.html#K12" title="SEQ_1:func.12">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E2:82_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:82"/>
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K12" title="SEQ_1:func.12">(#)</a> <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D9" target="_self" title="RFUNCT_1:def.9">Def9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_1.html#D11" target="_self" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a><br/>

<b>take </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_1.html#D10" target="_self" title="RFUNCT_1:def.10">RFUNCT_1:def 10</a><br/>

<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="E5:82"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seq_1.html#K12" title="SEQ_1:func.12">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E6:82"/><b>then </b><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seq_1.html#K12" title="SEQ_1:func.12">(#)</a> <font color="Maroon" title="c3">f</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="E7:82"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><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></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:82"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:82"/>
<font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seq_1.html#K12" title="SEQ_1:func.12">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E6:82"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>


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