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<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> ; <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">C</font> being 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/> for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b1">C</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> ( ( <font color="Olive" title="b2">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="Olive" title="b3">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="Olive" title="b2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b3">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> ) &amp; ( <font color="Olive" title="b2">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="Olive" title="b3">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="Olive" title="b2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b3">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> ) &amp; ( <font color="Olive" title="b2">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="Olive" title="b3">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="Olive" title="b2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b3">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> &amp; <font color="Olive" title="b3">f2</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">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> &amp; <font color="Olive" title="b2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b3">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> ) ) )</span><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> ; <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">f1</font>, <font color="Olive" title="b2">f2</font> being  <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>  holds <br/> ( ( <font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b1">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">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> ) &amp; ( <font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b1">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">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> ) &amp; ( <font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b1">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">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> &amp; <font color="Olive" title="b2">f2</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b2">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> ) ) )</span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <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="valued_1.html#K47" title="VALUED_1:func.47">-</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> ) &amp; ( <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="valued_1.html#K47" title="VALUED_1:func.47">-</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> ) &amp; ( <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> implies ( <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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> &amp; <font color="Maroon" title="c5">f2</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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> ) ) )</span><br/>









<b>thus </b><a NAME="E1:92"/>
( <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="valued_1.html#K47" title="VALUED_1:func.47">-</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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <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="valued_1.html#K47" title="VALUED_1:func.47">-</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> ) &amp; ( <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> implies ( <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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> &amp; <font color="Maroon" title="c5">f2</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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> ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1"/><i><font color="Green" title="E52">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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></span><br/>

<a NAME="E2:92_1"/><b>then </b>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</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/>
<b>hence </b><a NAME="E3:92_1"/>
<font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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="txt" href="rfunct_1.html#E1:92_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T110" target="_self" title="RFUNCT_1:th.110">Th110</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:92"/>
( <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="valued_1.html#K47" title="VALUED_1:func.47">-</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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies ( <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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> &amp; <font color="Maroon" title="c5">f2</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_2"/><i><font color="Green" title="E52">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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></span><br/>

<a NAME="E2:92_2"/><b>then </b>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</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/>
<b>hence </b><a NAME="E3:92_2"/>
<font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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="txt" href="rfunct_1.html#E1:92_2"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="rfunct_1.html#T110" target="_self" title="RFUNCT_1:th.110">Th110</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E3:92"/><i><font color="Green" title="E52">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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> &amp; <font color="Maroon" title="c5">f2</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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> )</span><br/>

<a NAME="E4:92"/><b>then </b>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</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/>
<b>hence </b><a NAME="E5:92"/>
<font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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#E3:92"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="rfunct_1.html#T110" target="_self" title="RFUNCT_1:th.110">Th110</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="c5">f2</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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> )</span><br/>

<a NAME="E6:92"/><i><font color="Green" title="E53">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#E3:92"><i><font color="Green" title="E52">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="E7:92"/>
<font color="Maroon" title="c5">f2</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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#E3:92"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="rfunct_1.html#T101" target="_self" title="RFUNCT_1:th.101">Th101</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="c4">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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></span><br/>

<b>thus </b><a NAME="E8:92"/>
<font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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#E3:92"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="rfunct_1.html#E6:92"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="rfunct_1.html#T101" target="_self" title="RFUNCT_1:th.101">Th101</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
