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<span class="kw">set </span><font color="Maroon" title="c5">C</font> =  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c7">f2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</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"> ( (  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b4">b</font></span>)</span>,<font color="Maroon" title="c4">M</font>) ) &amp; (  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b4">b</font></span>)</span>,<font color="Maroon" title="c4">M</font>) ) implies <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67_1_2"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b4">b</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">and </span><br/><a NAME="E2:67_1_2"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b4">b</font></span>)</span>,<font color="Maroon" title="c4">M</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="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_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">A</font>, <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font> holds  <font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c9">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</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="c6">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">B</font>)</span><br/><a NAME="E1:67_1_2_1"/>
<font color="Maroon" title="c8">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E3:67_1_2_1"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c10">a1</font>,<font color="Maroon" title="c4">M</font>) &amp; <font color="Maroon" title="c10">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> )
 ;<br/><a NAME="E4:67_1_2_1"/>
<font color="Maroon" title="c9">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">b1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E6:67_1_2_1"/><span class="lab"><font color="Green" title="E41">A13</font></span>: 
( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c4">M</font>) &amp; <font color="Maroon" title="c11">b1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> )
 ;<br/><a NAME="E7:67_1_2_1"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
<font color="Maroon" title="c11">b1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1_2_1"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E8:67_1_2_1"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c10">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_2_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E9:67_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">b1</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="txt" href="#E7:67_1_2_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:67_1_2_1"/>
<font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E8:67_1_2_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E7:67_1_2_1"><span class="lab"><font color="Green" title="E42">A14</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="E4:67_1_2"/>
<font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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"> verum</span><br/>


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