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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">f1</font>, <font color="Olive" title="b2">f2</font> being <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> ( ( <font color="Olive" title="b1">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Olive" title="b2">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> implies <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> ) &amp; ( <font color="Olive" title="b1">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp; <font color="Olive" title="b2">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> implies <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> ) &amp; ( <font color="Olive" title="b1">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Olive" title="b2">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> implies <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> ) &amp; ( <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> ) &amp; ( <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> ) )</span><br/>







<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T60" target="_self" title="RFUNCT_1:th.60">Th60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></i>
;<br/>
<b>hence </b><a NAME="E2:97"/>
( ( <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> ) &amp; ( <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> ) &amp; ( <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><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>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> ) )
 ; <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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