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<span class="kw">let </span><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<font color="Maroon" title="c1">D</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>)</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"> ( (  for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being    <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">D</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>) holds  <font color="Maroon" title="c2">o1</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">F1</font> <a href="lpspacc1.html#K1" title="LPSPACC1:func.1">+</a> <font color="Olive" title="b2">F2</font> ) &amp; (  for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being    <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">D</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>) holds  <font color="Maroon" title="c3">o2</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">F1</font> <a href="lpspacc1.html#K1" title="LPSPACC1:func.1">+</a> <font color="Olive" title="b2">F2</font> ) implies <font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11_1_2"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being    <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">D</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>) holds  <font color="Maroon" title="c2">o1</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">f1</font> <a href="lpspacc1.html#K1" title="LPSPACC1:func.1">+</a> <font color="Olive" title="b2">f2</font>
 <span class="kw">and </span><br/><a NAME="E2:11_1_2"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being    <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">D</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>) holds  <font color="Maroon" title="c3">o2</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">f1</font> <a href="lpspacc1.html#K1" title="LPSPACC1:func.1">+</a> <font color="Olive" title="b2">f2</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="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="11_1_2_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">f1</font>, <font color="Olive" title="b2">f2</font> being    <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">D</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>) holds  <font color="Maroon" title="c2">o1</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">o2</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</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">D</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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"> <font color="Maroon" title="c2">o1</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">o2</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font>)</span><br/><a NAME="E1:11_1_2_1"/>
<font color="Maroon" title="c2">o1</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f1</font> <a href="lpspacc1.html#K1" title="LPSPACC1:func.1">+</a> <font color="Maroon" title="c5">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_2"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:11_1_2_1"/>
<font color="Maroon" title="c2">o1</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">o2</font> <a href="lpspace1.html#K2" title="LPSPACE1:func.2">.</a> (<font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1_2"><span class="lab"><font color="Green" title="E7">A2</font></span></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="E4:11_1_2"/>
<font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</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"> verum</span><br/>


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