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<b>let </b><font color="Maroon" title="c1">seq</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"> for <font color="Olive" title="b1">h</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/> for <font color="Olive" title="b2">Ns</font> being <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">h</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b2">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b2">Ns</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Ns</font> being <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">Ns</font> be  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font>
 ; <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="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span><br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
 <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cfcont_1.html#T23" target="_self" title="CFCONT_1:th.23">Th23</a></i>;<br/>
<a NAME="E3:24"/>
 <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="cfcont_1.html#T24" target="_self" title="CFCONT_1:th.24">Th24</a></i>;<br/>
<b>hence </b><a NAME="E4:24"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">h</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:24"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="comseq_3.html#T14" title="COMSEQ_3:th.14">COMSEQ_3:14</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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