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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <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>   st <font color="Maroon" title="c1">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">h</font> holds <br/>( <font color="Olive" title="b1">h</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="rfunct_2.html#R5" title="RFUNCT_2:pred.5">is_monotone_on</a> <font color="Maroon" title="c1">Y</font> )</span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font> implies ( <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <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> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R5" title="RFUNCT_2:pred.5">is_monotone_on</a> <font color="Maroon" title="c1">Y</font> ) )</span><br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E27">A1</font></i>: 
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <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> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R5" title="RFUNCT_2:pred.5">is_monotone_on</a> <font color="Maroon" title="c1">Y</font> )</span><br/>

<a NAME="E2:50"/><b>then </b>
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <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>   st <font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/><font color="Maroon" title="c2">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">r2</font>
 
;<br/>
<b>hence </b><a NAME="E3:50"/>
<font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#D2" target="_self" title="RFUNCT_2:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R5" title="RFUNCT_2:pred.5">is_monotone_on</a> <font color="Maroon" title="c1">Y</font> )</span><br/>

<a NAME="E4:50"/>
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <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>   st <font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/><font color="Maroon" title="c2">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">r1</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:50"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:50"/>
<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>
 <b>by </b><i><a class="ref" href="rfunct_2.html#D3" target="_self" title="RFUNCT_2:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R5" title="RFUNCT_2:pred.5">is_monotone_on</a> <font color="Maroon" title="c1">Y</font> )</span><br/>

<a NAME="E6:50"/>
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <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>   st <font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/><font color="Maroon" title="c2">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">r1</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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">r2</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:50"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:50"/>
<font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#D4" target="_self" title="RFUNCT_2:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R5" title="RFUNCT_2:pred.5">is_monotone_on</a> <font color="Maroon" title="c1">Y</font> )</span><br/>

<a NAME="E8:50"/>
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <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>   st <font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/><font color="Maroon" title="c2">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">r1</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:50"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:50"/>
<font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#D5" target="_self" title="RFUNCT_2:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R5" title="RFUNCT_2:pred.5">is_monotone_on</a> <font color="Maroon" title="c1">Y</font></span><br/>

<b>hence </b><a NAME="E10:50"/>
<font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#R5" title="RFUNCT_2:pred.5">is_monotone_on</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#D6" target="_self" title="RFUNCT_2:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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