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<b>let </b><font color="Maroon" title="c1">a</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c2">fp</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:55"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">fp</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">fs1</font>, <font color="Maroon" title="c4">fs2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:55"/><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c2">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fs1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">fs2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fs2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E38"><i><font color="Green" title="E31">Lm15</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1"/>
( <font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><b>suppose </b></a><a NAME="E1:55_1_1"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c5">fs2</font> = <font color="Maroon" title="c4">fs2</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="wsierp_1.html#E3:55"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/><a NAME="E3:55_1_1"/>
<font color="Maroon" title="c3">fs1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:55"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">fs1</font> = <font color="Maroon" title="c3">fs1</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/> <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c2">fp</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">fs1</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c5">fs2</font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E3:55"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="rvsum_1.html#T127" title="RVSUM_1:th.127">RVSUM_1:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c6">fs1</font></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#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c5">fs2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T126" title="RVSUM_1:th.126">RVSUM_1:126</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c6">fs1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c5">fs2</font></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E6:55_1_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span> is    <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>by </b><i><a class="txt" href="wsierp_1.html#E1:55_1_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2"><b>suppose </b></a><a NAME="E1:55_1_2"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:55_1_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E3:55_1_2"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span> is    <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> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
