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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">V</font> being  <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a><br/> for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font>, <font color="Olive" title="b4">f3</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">V</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Olive" title="b3">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Olive" title="b4">f3</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Olive" title="b4">f3</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">V</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">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Olive" title="b3">f3</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Olive" title="b3">f3</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font>, <font color="Maroon" title="c5">f3</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</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="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span></span><br/>









<i><font color="Green" title="E7">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">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</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">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span></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="c5">f3</font></span>)</span>
<b>by </b><i><a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span></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="c5">f3</font></span>)</span>
<b>by </b><i><a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</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="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</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="c3">f1</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</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="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</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">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> )</span><br/><b>assume </b><a NAME="E1:9_2"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c6">x</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">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</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">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span><br/><a NAME="E2:9_2"/><b>then </b>
<font color="Maroon" title="c6">x</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> <font color="Maroon" title="c3">f1</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</a></i>;<br/><a NAME="E3:9_2"/><b>then </b><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c6">x</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">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:9_2"/>
<font color="Maroon" title="c6">x</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">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span></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="c5">f3</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:9_2"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</a></i>;<br/><a NAME="E5:9_2"/><b>then </b><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c6">x</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">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</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="c5">f3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:9_2"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</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="c5">f3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:9_2"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</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">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</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">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:9_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E1:9_2"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def1</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="E3:9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f3</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="partfun2.html#T3" title="PARTFUN2:th.3">PARTFUN2:3</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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