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<span class="kw">let </span><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">N</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</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"> (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  for <font color="Olive" title="b1">c</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">N</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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">c</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">N</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">c</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="c4">N</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">X</font> implies  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:21"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c4">N</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">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">  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span></span><br/>

<a NAME="E3:21"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="nat_1.html#T45" onmouseover="rs('nat_1/T45')" onmouseout="rh()">NAT_1:45</a></span>;<br/>
<span class="kw">thus </span> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">N</font> = 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">N</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="afinsq_2.html#T65" onmouseover="rs('afinsq_2/T65')" onmouseout="rh()">AFINSQ_2:65</a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="afinsq_2.html#T64" onmouseover="rs('afinsq_2/T64')" onmouseout="rh()">AFINSQ_2:64</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/>


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