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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="integra1.html#V1" target="_self">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:19"/><i><font color="Green">E15</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E5:19"/><i><font color="Green">E16</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:19"><i><font color="Green">E13</font></i></a>, <a class="txt" href="integra1.html#E4:19"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:19"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:19"><i><font color="Green">E13</font></i></a>, <a class="txt" href="integra1.html#E2:19"><i><font color="Green">E14</font></i></a>, <a class="txt" href="integra1.html#E4:19"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:19"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<a NAME="E8:19"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/>
<a NAME="E9:19"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E10:19"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K10">-</a> 0
 
<b>by </b><i><a class="txt" href="integra1.html#E5:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:19"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E8:19"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E13:19"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:19"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E14:19"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="integra1.html#E9:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="integra1.html#E10:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:19"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:19"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E17:19"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra1.html#E4:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="integra1.html#E9:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="integra1.html#E10:19"><i><font color="Green">E19</font></i></a>, <a class="txt" href="integra1.html#E13:19"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra1.html#E15:19"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


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