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<b>let </b><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">h1</font>, <font color="Maroon" title="c4">h2</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>;<br/><b>let </b><font color="Maroon" title="c5">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">CNS</font>;<br/>







<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E58">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span>
 ;<br/>

<a NAME="E2:64"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> <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="c3">h1</font> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> <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="c4">h2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:64"/><b>then </b><i><font color="Green" title="E59">A2</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">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="c3">h1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">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="c4">h2</font> )
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:64"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:64"/><i><font color="Green" title="E60">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:64"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>;<br/>
<a NAME="E5:64"/><i><font color="Green" title="E61">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:64"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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> ;<br/><a NAME="E1:64_1"/><i><font color="Green" title="E62">A5</font></i>: 
<font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E4:64"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="ncfcont1.html#T6" target="_self" title="NCFCONT1:th.6">Th6</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E4:64"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="ncfcont1.html#T11" target="_self" title="NCFCONT1:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:64_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:64"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="ncfcont1.html#T11" target="_self" title="NCFCONT1:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:64"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="ncfcont1.html#T11" target="_self" title="NCFCONT1:th.11">Th11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:64"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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> ;<br/><a NAME="E1:64_2"/><i><font color="Green" title="E62">A6</font></i>: 
<font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E5:64"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="ncfcont1.html#T6" target="_self" title="NCFCONT1:th.6">Th6</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E5:64"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="ncfcont1.html#T11" target="_self" title="NCFCONT1:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:64_2"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:64"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="ncfcont1.html#T11" target="_self" title="NCFCONT1:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:64"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="ncfcont1.html#T11" target="_self" title="NCFCONT1:th.11">Th11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:64"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c5">seq</font></span>)</span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>;<br/>


</div>
