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<span class="kw">let </span><font color="Maroon" title="c1">X</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Olive" title="b1">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b2">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  holds <br/><span class="p1">(<span class="default"><a href="mesfun7c.html#K5" title="MESFUN7C:func.5">superior_realsequence</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  holds <br/><span class="p1">(<span class="default"><a href="mesfun7c.html#K5" title="MESFUN7C:func.5">superior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  implies <span class="p1">(<span class="default"><a href="mesfun7c.html#K5" title="MESFUN7C:func.5">superior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c3">x</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  or <span class="p1">(<span class="default"><a href="mesfun7c.html#K5" title="MESFUN7C:func.5">superior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c3">x</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:30"/>
<font color="Maroon" title="c2">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfun7c.html#K5" title="MESFUN7C:func.5">superior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c3">x</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> </span><br/>

<a NAME="E3:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup1.html#T90" title="RINFSUP1:th.90">RINFSUP1:90</a></span>;<br/>
<a NAME="E4:30"/><span class="kw">then </span>
 <a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T9" title="RINFSUP2:th.9">RINFSUP2:9</a></span>;<br/>
<a NAME="E5:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<span class="p1">(<span class="default"><a href="mesfun7c.html#K5" title="MESFUN7C:func.5">superior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c3">x</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="mesfun7c.html#T11" target="_self" title="MESFUN7C:th.11">Th11</a></span>;<br/>
<a NAME="E6:30"/>
 <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font></span>)</span> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T58" title="RINFSUP1:th.58">RINFSUP1:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:30"/>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K5" title="MESFUN7C:func.5">superior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c3">x</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="rinfsup2.html#T13" title="RINFSUP2:th.13">RINFSUP2:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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