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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:15"/><i><font color="Green">E9</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<i><font color="Green">E10</font></i>: <span class="p1">(<span class="default"><a href="pralg_2.html#K14">OPS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D20">PRALG_2:def 20</a></i>
<br/>.= 
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:15"><i><font color="Green">E9</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E3:15"/><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_3.html#E1:15"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pralg_3.html#T8" target="_self">Th8</a></i>;<br/>
<i><font color="Green">E12</font></i>:  <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 

<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="pralg_3.html#E2:15"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p4">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 

;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:15"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E7:15"/><i><font color="Green">E14</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> </span>)</span> )
 <b>by </b><i><a class="txt" href="pralg_3.html#E3:15"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E8:15"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_3.html#E7:15"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:15"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E11:15"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="pralg_3.html#E6:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="pralg_3.html#E7:15"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E12:15"/><i><font color="Green">E16</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="pralg_2.html#T14">PRALG_2:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="pralg_3.html#E10:15"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="pralg_3.html#E3:15"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pralg_3.html#E4:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pralg_3.html#E10:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pralg_3.html#E12:15"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>
;<br/>
<b>hence </b><a NAME="E14:15"/>
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


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