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<span class="kw">let </span><font color="Maroon" title="c1">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">seq1</font>, <font color="Olive" title="b3">seq2</font> being  <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">N</font> holds  <font color="Olive" title="b1">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">seq1</font> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <font color="Olive" title="b3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Olive" title="b2">seq1</font></span>)</span> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Olive" title="b3">seq2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font> being  <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">N</font> holds  <font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <font color="Olive" title="b2">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Olive" title="b1">seq1</font></span>)</span> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Olive" title="b2">seq2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">seq1</font>, <font color="Maroon" title="c4">seq2</font> be   <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span><br/>







<span class="kw">thus </span><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <font color="Maroon" title="c4">seq2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <span class="p2">(<span class="default"><a href="toprns_1.html#K3" title="TOPRNS_1:func.3">-</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T13" target="_self" title="TOPRNS_1:th.13">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T12" target="_self" title="TOPRNS_1:th.12">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c4">seq2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T14" target="_self" title="TOPRNS_1:th.14">Th14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T14" target="_self" title="TOPRNS_1:th.14">Th14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c4">seq2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T12" target="_self" title="TOPRNS_1:th.12">Th12</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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