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<b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="mesfunc8.html#M1" title="MESFUNC8:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c2">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/>
<a NAME="E2:18"/><i><font color="Green" title="E13">P1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D5" target="_self" title="MESFUNC8:def.5">definf</a></i>;<br/>
<a NAME="E3:18"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K1" title="MESFUNC8:func.1">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D3" target="_self" title="MESFUNC8:def.3">definff</a></i>;<br/>
<a NAME="E4:18"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K1" title="MESFUNC8:func.1">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="kurato_2.html#D2" title="KURATO_2:def.2">KURATO_2:def 2</a></i>;<br/>
<a NAME="E5:18"/><b>then </b><i><font color="Green" title="E14">P2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K1" title="MESFUNC8:func.1">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D2" target="_self" title="MESFUNC8:def.2">defsamedom</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E15">AS1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K1" title="MESFUNC8:func.1">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:18_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/><a NAME="E2:18_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#D2" title="KURATO_2:def.2">KURATO_2:def 2</a></i>;<br/><a NAME="E3:18_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#D2" title="KURATO_2:def.2">KURATO_2:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E3:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/><a NAME="E4:18_1"/><b>then </b><i><font color="Green" title="E16">P3</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc8.html#K1" title="MESFUNC8:func.1">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc8.html#D3" target="_self" title="MESFUNC8:def.3">definff</a>, <a class="txt" href="mesfunc8.html#E1:18_1"><i><font color="Green" title="E15">AS1</font></i></a></i>;<br/><a NAME="E5:18_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="mesfunc8.html#D5" target="_self" title="MESFUNC8:def.5">definf</a>, <a class="txt" href="mesfunc8.html#E1:18_1"><i><font color="Green" title="E15">AS1</font></i></a>, <a class="txt" href="mesfunc8.html#E2:18"><i><font color="Green" title="E13">P1</font></i></a>, <a class="txt" href="mesfunc8.html#E5:18"><i><font color="Green" title="E14">P2</font></i></a></i>;<br/><b>hence </b><a NAME="E6:18_1"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K1" title="MESFUNC8:func.1">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E4:18_1"><i><font color="Green" title="E16">P3</font></i></a>, <a class="ref" href="rinfsup2.html#T27" title="RINFSUP2:th.27">RINFSUP2:27</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:18"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc8.html#K1" title="MESFUNC8:func.1">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E2:18"><i><font color="Green" title="E13">P1</font></i></a>, <a class="txt" href="mesfunc8.html#E5:18"><i><font color="Green" title="E14">P2</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


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