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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">f</font> being   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><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><br/>  for <font color="Olive" title="b4">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b3">F</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Olive" title="b4">r</font>)</span>)</span> ) holds <br/> for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Olive" title="b4">r</font>)</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">f</font> being   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><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><br/>  for <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b2">F</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Olive" title="b3">r</font>)</span>)</span> ) holds <br/> for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Olive" title="b3">r</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</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">F</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">F</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Olive" title="b2">r</font>)</span>)</span> ) holds <br/> for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="setlim_1.html#K5" title="SETLIM_1:func.5">inferior_setsequence</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Olive" title="b2">r</font>)</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">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c4">F</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Olive" title="b1">r</font>)</span>)</span> ) holds <br/> for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <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="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Olive" title="b1">r</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c4">F</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span> ) implies  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <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="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span> )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">E</font> =  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:28"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 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="c4">F</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c5">r</font>)</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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <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="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <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="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">n9</font> = <font color="Maroon" title="c7">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c3">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c4">F</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font>;<br/>
<div><span class="lab"><font color="Green" title="E25">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">now </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">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">k9</font> = <font color="Maroon" title="c11">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E2:28_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">k9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" onmouseover="rs('nat_1/D3')" onmouseout="rh()">NAT_1:def 3</a></span>;<br/><a NAME="E3:28_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:28_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" onmouseover="rs('nat_1/D3')" onmouseout="rh()">NAT_1:def 3</a></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/></div><span class="kw">end;</span></div>
<a NAME="E4:28"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc8.html#T2" target="_self" onmouseover="rs('mesfunc8/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">g</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E6:28"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">and </span><br/><a NAME="E7:28"/>
<font color="Maroon" title="c3">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">g</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font>
 ;<br/>
<a NAME="E8:28"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">g</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="nat_1.html#D3" onmouseover="rs('nat_1/D3')" onmouseout="rh()">NAT_1:def 3</a></span>;<br/>
<a NAME="E9:28"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="mesfunc8.html#D2" target="_self" onmouseover="rs('mesfunc8/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E10:28"/><span class="kw">then </span>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><a href="mesfunc8.html#K1" title="MESFUNC8:func.1">inf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c8">n9</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="mesfunc8.html#T16" target="_self" onmouseover="rs('mesfunc8/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:28"/>
<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="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<font color="Maroon" title="c5">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="mesfunc8.html#T8" target="_self" onmouseover="rs('mesfunc8/T8')" onmouseout="rh()">Th8</a></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/>


</div>
