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<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>









<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="msualg_4.html#NM3">ManySortedRelation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:83"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] )
 <b>from </b><i><a class="ref" href="msualg_6.html#S2" target="_self">MSUALG_6:sch 2</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msualg_6.html#NM4">EquationalTheory</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msualg_6.html#E2:83"><i><font color="Green">E56</font></i></a>, <a class="ref" href="msualg_6.html#T50" target="_self">Th50</a></i>;<br/>
<a NAME="E5:83"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:83_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:83_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:83_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a>  <a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E4:83_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E5:83_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:83_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rewrite1.html#T30">REWRITE1:30</a></i>;<br/>
<b>hence </b><a NAME="E6:83_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:83"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:83_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:83"/><b>then </b>
 <a href="msualg_6.html#K10" target="_self">EqTh</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E7:83"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_6.html#K10" target="_self">EqTh</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:83"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K10" target="_self">EqTh</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:83"><i><font color="Green">E56</font></i></a></i>;<br/>

<a NAME="E9:83"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a>  <a href="msualg_6.html#K10" target="_self">EqTh</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E10:83"/><b>then </b>
 <a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a>  <a href="msualg_6.html#K10" target="_self">EqTh</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E11:83"/>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K10" target="_self">EqTh</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="msualg_6.html#T51" target="_self">Th51</a></i>;<br/>


</div>
