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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> ; <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">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">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Olive" title="b2">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Olive" title="b2">n</font></span>)</span></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Olive" title="b1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Olive" title="b1">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">m</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c4">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c4">m</font></span><br/><a NAME="E1:15_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c4">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</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#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></i>;<br/><b>hence </b><a NAME="E2:15_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c4">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <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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