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

<span class="kw">let </span><font color="Maroon" title="c1">X1</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">X2</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X2</font> &amp; <font color="Olive" title="b1">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">X2</font> holds <br/> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b1">X2</font></span>)</span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">X1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">X2</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">X2</font> &amp; <font color="Olive" title="b2">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">X2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b1">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b2">X2</font></span>)</span>;<br/>
<a NAME="E1:50"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_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">X1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">X2</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">X2</font> &amp; <font color="Olive" title="b2">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">X2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b1">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b2">X2</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">X1</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">X2</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X2</font> &amp; <font color="Olive" title="b1">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">X2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">X2</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_1_1"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<font color="Maroon" title="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">and </span><br/><a NAME="E2:50_1_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c4">X2</font> <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">and </span><br/><a NAME="E3:50_1_1"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
 not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <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">  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><br/><span class="kw">set </span><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">X2</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c4">X2</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">m</font> =  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">X11</font> = <font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">X12</font> = <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E6:50_1_1"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D8" onmouseover="rs('xxreal_2/D8')" onmouseout="rh()">XXREAL_2:def 8</a></span>;<br/><a NAME="E7:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T116" onmouseover="rs('zfmisc_1/T116')" onmouseout="rh()">ZFMISC_1:116</a></span>;<br/><span class="lab"><font color="Green" title="E55">A9</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T49" onmouseover="rs('xboole_1/T49')" onmouseout="rh()">XBOOLE_1:49</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" onmouseover="rs('zfmisc_1/T46')" onmouseout="rh()">ZFMISC_1:46</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" onmouseover="rs('xboole_1/T37')" onmouseout="rh()">XBOOLE_1:37</a></span>
;<br/><a NAME="E9:50_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T45" onmouseover="rs('card_2/T45')" onmouseout="rh()">CARD_2:45</a></span>;<br/><a NAME="E10:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A10</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">X2</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="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_1"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E8:50_1_1"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">m</font> =  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E6:50_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a></span>;<br/><a NAME="E12:50_1_1"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
<font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a></span>;<br/><a NAME="E13:50_1_1"/><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/><a NAME="E14:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A13</font></span>: 
( <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E3:50_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a></span>;<br/><a NAME="E15:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A14</font></span>: 
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E10:50_1_1"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_1_2"><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> <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c12">r</font> = <font color="Maroon" title="c11">x</font>;<br/><span class="kw">assume </span><a NAME="E1:50_1_1_2"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font></span><br/><a NAME="E2:50_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:50_1_1"><span class="lab"><font color="Green" title="E58">A12</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">r</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a></span>;<br/><a NAME="E4:50_1_1_2"/>
 not <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="#E13:50_1_1"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E1:50_1_1_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a></span>;<br/><a NAME="E5:50_1_1_2"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E6:50_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A16</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E7:50_1_1_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:50_1_1"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E1:50_1_1_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="xxreal_2.html#D8" onmouseover="rs('xxreal_2/D8')" onmouseout="rh()">XXREAL_2:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E8:50_1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</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><a NAME="E17:50_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font>
 ;<br/><a NAME="E18:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">a17</font></span>: 
<font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="finseq_1.html#V3" title="FINSEQ_1:attr.3">included_in_Seg</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" onmouseover="rs('finseq_1/D13')" onmouseout="rh()">FINSEQ_1:def 13</a></span>;<br/><a NAME="E19:50_1_1"/><span class="lab"><font color="Green" title="E62">A18</font></span>: 
 not <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="xxreal_2.html#D8" onmouseover="rs('xxreal_2/D8')" onmouseout="rh()">XXREAL_2:def 8</a></span>;<br/><a NAME="E20:50_1_1"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E21:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A19</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E22:50_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E23:50_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E24:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">a20</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="finseq_1.html#V3" title="FINSEQ_1:attr.3">included_in_Seg</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" onmouseover="rs('finseq_1/D13')" onmouseout="rh()">FINSEQ_1:def 13</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_1_3"><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">n1</font>, <font color="Olive" title="b2">n2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">n2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">n1</font>, <font color="Maroon" title="c12">n2</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c11">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">n2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_1_1_3"/><span class="lab"><font color="Green" title="E65">A21</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E2:50_1_1_3"/><span class="lab"><font color="Green" title="E66">A22</font></span>: 
<font color="Maroon" title="c12">n2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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"> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n2</font></span><br/><a NAME="E3:50_1_1_3"/>
 not <font color="Maroon" title="c11">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1_3"><span class="lab"><font color="Green" title="E65">A21</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:50_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A23</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:50_1_1_3"/>
( <font color="Maroon" title="c12">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> &amp; <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:50_1_1"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E1:50_1_1_3"><span class="lab"><font color="Green" title="E65">A21</font></span></a>, <a class="txt" href="#E2:50_1_1_3"><span class="lab"><font color="Green" title="E66">A22</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xxreal_2.html#D8" onmouseover="rs('xxreal_2/D8')" onmouseout="rh()">XXREAL_2:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E6:50_1_1_3"/>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1_3"><span class="lab"><font color="Green" title="E67">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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><a NAME="E26:50_1_1"/><span class="kw">then </span>
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_1"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E18:50_1_1"><span class="lab"><font color="Green" title="E61">a17</font></span></a>, <a class="txt" href="#E24:50_1_1"><span class="lab"><font color="Green" title="E64">a20</font></span></a>, <a class="ref" href="finseq_3.html#T42" onmouseover="rs('finseq_3/T42')" onmouseout="rh()">FINSEQ_3:42</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A24</font></span>:  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T97" onmouseover="rs('rvsum_1/T97')" onmouseout="rh()">RVSUM_1:97</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:50_1_1"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="ref" href="finseq_3.html#T44" onmouseover="rs('finseq_3/T44')" onmouseout="rh()">FINSEQ_3:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c10">m</font>

;<br/><a NAME="E28:50_1_1"/><span class="lab"><font color="Green" title="E66">A25</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:50_1_1"><span class="lab"><font color="Green" title="E63">A19</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_5"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1_1_5"/>
( <font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X1</font> or  not <font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_5_1"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_5_1"/><span class="lab"><font color="Green" title="E67">A26</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X1</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="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E68">A27</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_1_5_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">n1</font>, <font color="Olive" title="b2">n2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">n2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">n1</font>, <font color="Maroon" title="c12">n2</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c11">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">n2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_1_1_5_1_1"/><span class="lab"><font color="Green" title="E69">A28</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E2:50_1_1_5_1_1"/><span class="lab"><font color="Green" title="E70">A29</font></span>: 
<font color="Maroon" title="c12">n2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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"> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n2</font></span><br/><a NAME="E3:50_1_1_5_1_1"/>
 not <font color="Maroon" title="c11">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1_5_1_1"><span class="lab"><font color="Green" title="E69">A28</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:50_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A30</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:50_1_1_5_1_1"/>
<font color="Maroon" title="c11">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:50_1_1"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E1:50_1_1_5_1_1"><span class="lab"><font color="Green" title="E69">A28</font></span></a></span>;<br/><a NAME="E6:50_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A31</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:50_1_1"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="xxreal_2.html#D8" onmouseover="rs('xxreal_2/D8')" onmouseout="rh()">XXREAL_2:def 8</a></span>;<br/><a NAME="E7:50_1_1_5_1_1"/>
<font color="Maroon" title="c12">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1_5_1_1"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:50_1_1_5_1_1"/>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1_5_1_1"><span class="lab"><font color="Green" title="E71">A30</font></span></a>, <a class="txt" href="#E6:50_1_1_5_1_1"><span class="lab"><font color="Green" title="E72">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_1_5_1_2"><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> <font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c12">r</font> = <font color="Maroon" title="c11">x</font>;<br/><span class="kw">assume </span><a NAME="E1:50_1_1_5_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font></span><br/><a NAME="E2:50_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A32</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:50_1_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">r</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a></span>;<br/><a NAME="E5:50_1_1_5_1_2"/>
 not <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E3:50_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="#E2:50_1_1_5_1_2"><span class="lab"><font color="Green" title="E69">A32</font></span></a></span>;<br/><a NAME="E6:50_1_1_5_1_2"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E7:50_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A33</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:50_1_1_5_1_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E2:50_1_1_5_1_2"><span class="lab"><font color="Green" title="E69">A32</font></span></a>, <a class="ref" href="xxreal_2.html#D8" onmouseover="rs('xxreal_2/D8')" onmouseout="rh()">XXREAL_2:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E9:50_1_1_5_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_1_5_1_2"><span class="lab"><font color="Green" title="E70">A33</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</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><a NAME="E4:50_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">m</font>
 ;<br/><a NAME="E5:50_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">a34</font></span>: 
<font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="finseq_1.html#V3" title="FINSEQ_1:attr.3">included_in_Seg</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" onmouseover="rs('finseq_1/D13')" onmouseout="rh()">FINSEQ_1:def 13</a></span>;<br/><a NAME="E6:50_1_1_5_1"/>
<font color="Maroon" title="c3">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1_5_1"><span class="lab"><font color="Green" title="E67">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#T116" onmouseover="rs('zfmisc_1/T116')" onmouseout="rh()">ZFMISC_1:116</a></span>;<br/><span class="kw">then </span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c3">X1</font></span>)</span> = 
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:50_1_1"><span class="lab"><font color="Green" title="E64">a20</font></span></a>, <a class="txt" href="#E5:50_1_1_5_1"><span class="lab"><font color="Green" title="E69">a34</font></span></a>, <a class="txt" href="#E2:50_1_1_5_1"><span class="lab"><font color="Green" title="E68">A27</font></span></a>, <a class="ref" href="finseq_3.html#T42" onmouseover="rs('finseq_3/T42')" onmouseout="rh()">FINSEQ_3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T97" onmouseover="rs('rvsum_1/T97')" onmouseout="rh()">RVSUM_1:97</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">m</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:50_1_1"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="ref" href="finseq_3.html#T44" onmouseover="rs('finseq_3/T44')" onmouseout="rh()">FINSEQ_3:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c10">m</font>

;<br/><span class="kw">hence </span><a NAME="E8:50_1_1_5_1"/>
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E12:50_1_1"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E10:50_1_1"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E14:50_1_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E27:50_1_1"><span class="lab"><font color="Green" title="E65">A24</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_5_2"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_5_2"/>
 not <font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X1</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="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><br/><div class="add"><a NAME="E2:50_1_1_5_2"/><span class="kw">then </span>
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:50_1_1"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T57" onmouseover="rs('zfmisc_1/T57')" onmouseout="rh()">ZFMISC_1:57</a></span>;<br/><span class="kw">hence </span><a NAME="E3:50_1_1_5_2"/>
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:50_1_1"><span class="lab"><font color="Green" title="E65">A24</font></span></a>, <a class="txt" href="#E28:50_1_1"><span class="lab"><font color="Green" title="E66">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:50_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c2">X2</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X2</font> &amp; <font color="Maroon" title="c2">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> implies  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c2">X2</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:50"/><span class="lab"><font color="Green" title="E48">A35</font></span>: 
( <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X2</font> &amp; <font color="Maroon" title="c2">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</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="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c2">X2</font></span>)</span></span><br/>

<a NAME="E3:50"/><span class="lab"><font color="Green" title="E49">A36</font></span>: 
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font>
 
;<br/>
<a NAME="E4:50"/><span class="lab"><font color="Green" title="E50">A37</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">X1</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">X2</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X2</font> &amp; <font color="Olive" title="b1">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">X2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b1">X2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">X2</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X2</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_2"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
<font color="Maroon" title="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">and </span><br/><a NAME="E2:50_2"/>
<font color="Maroon" title="c4">X2</font> <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">and </span><br/><a NAME="E3:50_2"/>
 not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">and </span><br/><a NAME="E4:50_2"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><br/>

<a NAME="E5:50_2"/>
<font color="Maroon" title="c4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_2"><span class="lab"><font color="Green" title="E52">A39</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:50_2"/>
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="xboole_1.html#T3" onmouseover="rs('xboole_1/T3')" onmouseout="rh()">XBOOLE_1:3</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>
<a NAME="E5:50"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E47">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:50"/>
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c2">X2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E49">A36</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>
