<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  <a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">thus </span><a NAME="E1:21"/>
 <a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a>  <a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D7" onmouseover="rs('binop_1/D7')" onmouseout="rh()">BINOP_1:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a>  <a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">f</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D16" onmouseover="rs('binop_1/D16')" onmouseout="rh()">BINOP_1:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">h</font> = <span class="p1">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<span class="p1">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>);<br/>
<a NAME="E1:21_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T4" target="_self" onmouseover="rs('lpspacc1/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E2:21_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<a NAME="E3:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">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">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<span class="p1">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><span class="kw">assume </span><a NAME="E2:21_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="lpspacc1.html#T4" target="_self" onmouseover="rs('lpspacc1/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>

;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:21_1"/>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c2">f</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D17" onmouseover="rs('binop_1/D17')" onmouseout="rh()">BINOP_1:def 17</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">h</font> = <span class="p1">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>);<br/>
<a NAME="E2:21"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T4" target="_self" onmouseover="rs('lpspacc1/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E3:21"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<a NAME="E4:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><a NAME="E1:21_2"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<span class="p1">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><span class="kw">assume </span><a NAME="E2:21_2"/>
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="lpspacc1.html#T4" target="_self" onmouseover="rs('lpspacc1/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>

;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:21"/>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K9" title="LPSPACC1:func.9">addcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K10" title="LPSPACC1:func.10">CPFuncZero</a> <font color="Maroon" title="c1">A</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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