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<b>let </b><font color="Maroon" title="c1">D</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">G</font>, <font color="Olive" title="b2">H</font>, <font color="Olive" title="b3">J</font> being   <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Olive" title="b2">H</font></span>)</span> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Olive" title="b3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Olive" title="b3">J</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Olive" title="b3">J</font></span>)</span> &amp; <font color="Olive" title="b3">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Olive" title="b2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Olive" title="b2">H</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font>, <font color="Maroon" title="c4">J</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> &amp; <font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> )</span><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">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> ; <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"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</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">G</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D8" target="_self" title="SEQFUNC:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D6" target="_self" title="SEQFUNC:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T22" title="RFUNCT_1:th.22">RFUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D8" target="_self" title="SEQFUNC:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D8" target="_self" title="SEQFUNC:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D6" target="_self" title="SEQFUNC:def.6">Def6</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="E2:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span>
 <b>by </b><i><a class="ref" href="seqfunc.html#T2" target="_self" title="SEQFUNC:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span></span><br/>

<b>hence </b><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#T6" target="_self" title="SEQFUNC:th.6">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c4">J</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#T6" target="_self" title="SEQFUNC:th.6">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#T6" target="_self" title="SEQFUNC:th.6">Th6</a></i>
;<br/> <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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