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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>; <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">RNS</font> being  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/> for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Olive" title="b1">RNS</font> holds  <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">X</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span><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>; <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">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <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> holds  <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">X</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span><br/><b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">f</font> being  <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> holds  <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span><br/><b>let </b><font color="Maroon" title="c4">f</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span><br/>







<i><font color="Green" title="E103">A1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
<b>by </b><i><a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span>
<b>by </b><i><a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">c</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">c</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> )</span><br/><b>assume </b><a NAME="E1:136_2"/><i><font color="Green" title="E104">A2</font></i>: 
<font color="Maroon" title="c5">c</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 ; <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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font></span><br/><a NAME="E2:136_2"/><b>then </b>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E3:136_2"/><b>then </b><i><font color="Green" title="E105">A3</font></i>: 
( <font color="Maroon" title="c5">c</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"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:136_2"/><i><font color="Green" title="E106">A4</font></i>: 
<font color="Maroon" title="c5">c</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="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:136"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E1:136_2"><i><font color="Green" title="E104">A2</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> = 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:136_2"><i><font color="Green" title="E104">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">c</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:136_2"><i><font color="Green" title="E105">A3</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">c</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E4:136_2"><i><font color="Green" title="E106">A4</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:136"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E1:136_2"><i><font color="Green" title="E104">A2</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:136"/>
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:136"><i><font color="Green" title="E103">A1</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
