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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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">F</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">S</font>  ex <font color="Olive" title="b4">G</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b4">G</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Olive" title="b3">F</font> &amp; <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">G</font></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">F</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">S</font>  ex <font color="Olive" title="b3">G</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b3">G</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">G</font></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">S</font>  ex <font color="Olive" title="b2">G</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">G</font></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">G</font></span>)</span></span>)</span> )</span><br/>







<a NAME="E1:3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">G</font> =  <a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Maroon" title="c4">F</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" onmouseover="rs('funct_2/T6')" onmouseout="rh()">FUNCT_2:6</a></span>;<br/>
<a NAME="E3:3"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c5">G</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D5" onmouseover="rs('prob_1/D5')" onmouseout="rh()">PROB_1:def 5</a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/>
<a NAME="E4:3"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure2.html#T23" onmouseover="rs('measure2/T23')" onmouseout="rh()">MEASURE2:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:3"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">G</font></span>)</span></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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