<?xml version="1.0"?>
<div class="add">

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="77_1_1_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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c1">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c1">f</font></span>)</span> implies <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c3">xx</font> = <font color="Maroon" title="c2">x</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:77_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c1">f</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"> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> </span><br/><a NAME="E3:77_1_1_1"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">xx</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#D8" target="_self" onmouseover="rs('flexary1/D8')" onmouseout="rh()">Def8</a></span>;<br/><a NAME="E4:77_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">xx</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D6" onmouseover="rs('valued_0/D6')" onmouseout="rh()">VALUED_0:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">fx</font> = <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/><a NAME="E6:77_1_1_1"/>
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <font color="Maroon" title="c4">fx</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>
 ;<br/><span class="kw">hence </span><a NAME="E7:77_1_1_1"/>
<span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></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>
<span class="kw">hence </span><a NAME="E2:77_1_1"/>
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c1">f</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D12" onmouseover="rs('valued_0/D12')" onmouseout="rh()">VALUED_0:def 12</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>
