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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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/> for <font color="Olive" title="b2">z</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font> holds   <a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="seq_4.html#K20" title="SEQ_4:func.20">*</a> <font color="Olive" title="b2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Olive" title="b1">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Olive" title="b2">z</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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">z</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font> holds   <a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="seq_4.html#K20" title="SEQ_4:func.20">*</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Olive" title="b1">z</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <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">  <a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="seq_4.html#K20" title="SEQ_4:func.20">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="124_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <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="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="seq_4.html#K20" title="SEQ_4:func.20">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">w</font> = <font color="Maroon" title="c4">j</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:124_1"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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"> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="seq_4.html#K20" title="SEQ_4:func.20">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">c9</font> = <font color="Maroon" title="c3">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>, <font color="Maroon" title="c7">cc</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="seq_4.html#K20" title="SEQ_4:func.20">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T74" target="_self" title="SEQ_4:th.74">Th21</a></span>;<br/><a NAME="E4:124_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/><a NAME="E5:124_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">ac</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">w</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:124_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="seq_4.html#K20" title="SEQ_4:func.20">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c7">cc</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:124_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="seq_4.html#T106" target="_self" title="SEQ_4:th.106">Th59</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c6">c9</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:124_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="seq_4.html#T99" target="_self" title="SEQ_4:th.99">Th52</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">c9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">ac</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:124_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="seq_4.html#T106" target="_self" title="SEQ_4:th.106">Th59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T67" title="RVSUM_1:th.67">RVSUM_1:67</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:124"/>
 <a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="seq_4.html#K20" title="SEQ_4:func.20">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T139" title="FINSEQ_2:th.139">FINSEQ_2:139</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/>


</div>
