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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">X2</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X2</font> &amp; <font color="Olive">X2</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp; not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Olive">X2</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X2</font></span>)</span>;<br/>
<a NAME="E1:50"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X2</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp; not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:50_1"/><b>then </b>
<font color="Maroon">X2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1"/>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:50"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X2</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp; not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 ;<br/><b>set </b><font color="Maroon">A</font> = <font color="Maroon">X2</font>;<br/><div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 ;<br/><a NAME="E2:50_2_1_1"/><b>then </b>
<font color="Maroon">y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:50_2_1_1"/>
<font color="Maroon">y</font> is <a href="xreal_0.html#V1">real</a>
 ;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">X2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V2">real-membered</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="txt" href="nat_4.html#E4:50_2_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="membered.html#D2">MEMBERED:def 2</a></i>;<br/><b>set </b><font color="Maroon">m</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font>;<br/><b>set </b><font color="Maroon">X11</font> = <font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><b>set </b><font color="Maroon">X12</font> = <font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E4:50_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a>, </i>;<br/><a NAME="E5:50_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E6:50_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, </i>;<br/><a NAME="E7:50_2_1"/><i><font color="Green">E56</font></i>: 
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E8:50_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T140">ZFMISC_1:140</a></i>;<br/><i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T52">ZFMISC_1:52</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
;<br/><a NAME="E10:50_2_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T64">CARD_2:64</a></i>;<br/><a NAME="E11:50_2_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E8:50_2_1"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E12:50_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 0
 <b>by </b><i/>;<br/><a NAME="E13:50_2_1"/><i><font color="Green">E62</font></i>: 
not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E6:50_2_1"><i><font color="Green">E55</font></i></a>, </i>;<br/><a NAME="E14:50_2_1"/><i><font color="Green">E63</font></i>: 
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E12:50_2_1"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="nat_4.html#E5:50_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_4.html#E7:50_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_4.html#E9:50_2_1"><i><font color="Green">E60</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_2_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:50_2_1_3"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E6:50_2_1"><i><font color="Green">E55</font></i></a></i>;<br/><b>set </b><font color="Maroon">r</font> = <font color="Maroon">x</font>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="nat_4.html#E1:50_2_1_3"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E4:50_2_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E14:50_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nat_4.html#E6:50_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E5:50_2_1_3"/>
not <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E14:50_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nat_4.html#E6:50_2_1"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E6:50_2_1_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E7:50_2_1_3"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:50_2_1_3"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E9:50_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><a NAME="E17:50_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E18:50_2_1"/><i><font color="Green">E70</font></i>: 
not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E19:50_2_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E20:50_2_1"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E21:50_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E22:50_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E23:50_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:50_2_1_4"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><a NAME="E2:50_2_1_4"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:50_2_1_4"/>
not <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:50_2_1_4"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:50_2_1_4"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a>, , <a class="txt" href="nat_4.html#E6:50_2_1"><i><font color="Green">E55</font></i></a></i>;<br/><b>hence </b><a NAME="E6:50_2_1_4"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E17:50_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/></div><b>end;</b></div><a NAME="E25:50_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E8:50_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="nat_4.html#E14:50_2_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><i><font color="Green">E77</font></i>:  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T127">RVSUM_1:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T50">FINSEQ_3:50</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T125">RVSUM_1:125</a></i>
;<br/><a NAME="E27:50_2_1"/><i><font color="Green">E78</font></i>: 
1 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_6"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_2_1_6"/>
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> or  not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_6_1"><b>suppose </b></a><a NAME="E1:50_2_1_6_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><div class="add"><a NAME="E2:50_2_1_6_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T140">ZFMISC_1:140</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_2_1_6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:50_2_1_6_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:50_2_1_6_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">r</font> = <font color="Maroon">x</font>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="nat_4.html#E23:50_2_1"><i><font color="Green">E72</font></i></a>, </i>;<br/><a NAME="E5:50_2_1_6_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:50_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E6:50_2_1_6_1_1"/>
not <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:50_2_1"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E7:50_2_1_6_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E8:50_2_1_6_1_1"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:50_2_1_6_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E10:50_2_1_6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><a NAME="E4:50_2_1_6_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_6_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:50_2_1_6_1_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><a NAME="E2:50_2_1_6_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:50_2_1_6_1_2"/>
not <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E23:50_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:50_2_1_6_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:50_2_1_6_1_2"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E23:50_2_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nat_4.html#E5:50_2_1"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E6:50_2_1_6_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E6:50_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>hence </b><a NAME="E7:50_2_1_6_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/></div><b>end;</b></div><b>then </b> <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X1</font></span>)</span> = 
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E18:50_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E21:50_2_1"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T127">RVSUM_1:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T50">FINSEQ_3:50</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="xboole_0.html#K4">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T125">RVSUM_1:125</a></i>
;<br/><b>hence </b><a NAME="E7:50_2_1_6_1"/>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X2</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E13:50_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_6_2"><b>suppose </b></a><a NAME="E1:50_2_1_6_2"/>
not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><div class="add"><a NAME="E2:50_2_1_6_2"/><b>then </b>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="xboole_0.html#K4">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E13:50_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><b>hence </b><a NAME="E3:50_2_1_6_2"/>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X2</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E17:50_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:50"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="nat_4.html#T4" target="_self">Th4</a>, );<br/></i>
<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X2</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E4:50"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp; not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 ;<br/>

<a NAME="E5:50"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="card_1.html#K4">card</a> <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font>
 
;<br/>
<b>hence </b><a NAME="E6:50"/>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X2</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
