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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">h</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="E1:52"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c1">Y</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">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#D5" target="_self" title="RFUNCT_2:def.5">RFUNCT_2:def 5</a><br/><b>let </b><font color="Maroon" title="c4">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="E2:52"/>
( <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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="c2">h</font></span>)</span> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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="c2">h</font></span>)</span> &amp; <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r2</font> )
 ;<br/>

<b>hence </b><a NAME="E3:52"/>
<font color="Maroon" title="c2">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">r1</font>
 <b>by </b><i><a class="txt" href="rfunct_2.html#E1:52"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="rfunct_2.html#D3" target="_self" title="RFUNCT_2:def.3">Def3</a></i>;<br/>


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