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<span class="kw">let </span><font color="Maroon" title="c1">CNS1</font>, <font color="Maroon" title="c2">CNS2</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</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">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">CNS1</font>,<font color="Maroon" title="c2">CNS2</font> holds  <span class="p1"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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#K1" title="NCFCONT1:func.1">||.</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#K1" title="NCFCONT1:func.1">.||</a></span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">CNS1</font>,<font color="Maroon" title="c2">CNS2</font> holds  <span class="p1"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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#K1" title="NCFCONT1:func.1">||.</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#K1" title="NCFCONT1:func.1">.||</a></span></span><br/><span class="kw">let </span><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">CNS1</font>,<font color="Maroon" title="c2">CNS2</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"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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#K1" title="NCFCONT1:func.1">||.</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#K1" title="NCFCONT1:func.1">.||</a></span></span><br/>







<span class="lab"><font color="Green" title="E77">A1</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ncfcont1.html#D2" target="_self" title="NCFCONT1:def.2">Def2</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</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#K1" title="NCFCONT1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ncfcont1.html#D2" target="_self" title="NCFCONT1:def.2">Def2</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">CNS1</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="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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#K1" title="NCFCONT1:func.1">||.</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#K1" title="NCFCONT1:func.1">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> )</span><br/><span class="kw">assume </span><a NAME="E1:129_2"/><span class="lab"><font color="Green" title="E78">A2</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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#K1" title="NCFCONT1:func.1">||.</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#K1" title="NCFCONT1:func.1">.||</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="E2:129_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A3</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="ncfcont1.html#D2" target="_self" title="NCFCONT1:def.2">Def2</a></span>;<br/><a NAME="E3:129_2"/>
<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#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129_2"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E4:129_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A4</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129_2"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</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="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129_2"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="ref" href="ncfcont1.html#D2" target="_self" title="NCFCONT1:def.2">Def2</a></span>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</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="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129_2"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></span>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</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#K1" title="NCFCONT1:func.1">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E1:129_2"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="ncfcont1.html#D2" target="_self" title="NCFCONT1:def.2">Def2</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="E3:129"/>
<span class="p1"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</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#K1" title="NCFCONT1:func.1">||.</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#K1" title="NCFCONT1:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</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>
