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<a NAME="E1:22_1_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:22_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><a NAME="E3:22_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T111">FINSEQ_2:111</a></i>;<br/><b>hence </b><a NAME="E4:22_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:22_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<a NAME="E3:22_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E4:22_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T111">FINSEQ_2:111</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="finseq_2.html#NM2">FinSequence-DOMAIN</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>thus </b><a NAME="E2:22_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:22_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>

<b>assume </b><a NAME="E3:22_1_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:22_1_1_2"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  ) &amp; ( <font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  ) &amp; ( <font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:22_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="chain_1.html#E3:22_1_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E2:22_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:22_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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