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<span class="kw">let </span><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq9</font>, <font color="Maroon" title="c4">seq19</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  &amp; <font color="Maroon" title="c3">seq9</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a> 
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c3">seq9</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c3">seq9</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c3">seq9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="seq_1.html#T43" target="_self" onmouseover="rs('seq_1/T43')" onmouseout="rh()">Th41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="seq_1.html#T43" target="_self" onmouseover="rs('seq_1/T43')" onmouseout="rh()">Th41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T15" target="_self" onmouseover="rs('seq_1/T15')" onmouseout="rh()">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c3">seq9</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c3">seq9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="seq_1.html#T43" target="_self" onmouseover="rs('seq_1/T43')" onmouseout="rh()">Th41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="seq_1.html#T43" target="_self" onmouseover="rs('seq_1/T43')" onmouseout="rh()">Th41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T20" target="_self" onmouseover="rs('seq_1/T20')" onmouseout="rh()">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq19</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq9</font></span>)</span>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
