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<b>let </b><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a> implies <font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c1">seq</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c1">seq</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c2">seq1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c2">seq1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="txt" href="comseq_1.html#E1:34"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="comseq_1.html#E2:34"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="comseq_1.html#T37" target="_self" title="COMSEQ_1:th.37">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="comseq_1.html#T11" target="_self" title="COMSEQ_1:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:34"/>
<font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c1">seq</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6" target="_self" title="COMSEQ_1:th.6">Th6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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