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<span class="kw">thus </span><a NAME="E1:19_1_1"/>
 <a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</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#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</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#D2" onmouseover="rs('binop_1/D2')" onmouseout="rh()">BINOP_1:def 2</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#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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="c3">g</font>,<font color="Maroon" title="c2">f</font>)</span><br/>

<a NAME="E1:19_1_1_1"/><span class="lab"><font color="Green" title="E10">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#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<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> <font color="Maroon" title="c3">g</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#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#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E2:19_1_1_1"/><span class="lab"><font color="Green" title="E11">A2</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#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</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> <font color="Maroon" title="c2">f</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_1_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> <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> <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="c3">g</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<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></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> <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> <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="c3">g</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<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> )</span><br/><span class="kw">assume </span><a NAME="E1:19_1_1_1_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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> <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="c3">g</font></span>)</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<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></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="#E2:19_1_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E1:19_1_1_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>
;<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="E4:19_1_1_1"/>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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="c3">g</font>,<font color="Maroon" title="c2">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E1:19_1_1_1"><span class="lab"><font color="Green" title="E10">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>, <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">h</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#D3" onmouseover="rs('binop_1/D3')" onmouseout="rh()">BINOP_1:def 3</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#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">j</font> = <span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">k</font> = <span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">j1</font> = <span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">k1</font> = <span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>);<br/>
<a NAME="E2:19_1_1"/><span class="lab"><font color="Green" title="E10">A4</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#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</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="c4">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E3:19_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A5</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#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E4:19_1_1"/><span class="lab"><font color="Green" title="E12">A6</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#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</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="c4">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E5:19_1_1"/><span class="lab"><font color="Green" title="E13">A7</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#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</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> <font color="Maroon" title="c2">f</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E6:19_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A8</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#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</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> <font color="Maroon" title="c2">f</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E7:19_1_1"/><span class="lab"><font color="Green" title="E15">A9</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#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_1_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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>)</span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>)</span>)</span> <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="c9">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="c9">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>)</span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:19_1_1_2"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>)</span>)</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><a NAME="E2:19_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19_1_1"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<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="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<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="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1_1"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="txt" href="#E1:19_1_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E2:19_1_1_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p3">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="lpspacc1.html#T5" target="_self" onmouseover="rs('lpspacc1/T5')" onmouseout="rh()">Th5</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">hence </span><a NAME="E9:19_1_1"/>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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="c3">g</font>,<font color="Maroon" title="c4">h</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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"><span class="p2">(<span class="default"><a href="lpspacc1.html#K7" title="LPSPACC1:func.7">multcpfunc</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>,<font color="Maroon" title="c3">g</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19_1_1"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</a>, <a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</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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