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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/>





<a NAME="E1:19"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D1" title="FINSEQ_5:def.1">FINSEQ_5:def 1</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:19"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E4:19"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:19"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E7:19"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="jordan1g.html#E3:19"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="finseq_5.html#T52" title="FINSEQ_5:th.52">FINSEQ_5:52</a></i>;<br/>
<a NAME="E8:19"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1g.html#E2:19"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan1g.html#E3:19"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan1g.html#E4:19"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="finseq_5.html#T49" title="FINSEQ_5:th.49">FINSEQ_5:49</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:19"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <b>and </b><br/><a NAME="E11:19"/><i><font color="Green" title="E20">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T52" title="FINSEQ_5:th.52">FINSEQ_5:52</a></i>;<br/>
<a NAME="E12:19"/><i><font color="Green" title="E21">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D1" title="FINSEQ_5:def.1">FINSEQ_5:def 1</a></i>;<br/>
<a NAME="E13:19"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan1g.html#E4:19"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan1g.html#E6:19"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E14:19"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1g.html#E2:19"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="finseq_6.html#T61" title="FINSEQ_6:th.61">FINSEQ_6:61</a></i>;<br/>
<b>then </b><i><font color="Green" title="E23">A11</font></i>: <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c5">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="jordan1g.html#E13:19"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/>
<a NAME="E16:19"/>
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan1g.html#E2:19"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan1g.html#E3:19"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan1g.html#E4:19"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="sprect_5.html#T3" title="SPRECT_5:th.3">SPRECT_5:3</a></i>;<br/>
<a NAME="E17:19"/><b>then </b>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font>
 
<b>by </b><i><a class="txt" href="jordan1g.html#E6:19"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="jordan1g.html#E10:19"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="jordan1g.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan1g.html#E7:19"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan1g.html#E11:19"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="jordan1g.html#E12:19"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="jordan1g.html#E15:19"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="jordan3.html#T21" title="JORDAN3:th.21">JORDAN3:21</a></i>;<br/>


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