<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>thus </b><a NAME="E1:13_1_1"/>
 <a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="altcat_1.html#V2">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>assume </b><a NAME="E2:13_1_1_1"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E3:13_1_1_1"/><b>then </b>
( the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<a NAME="E4:13_1_1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
(  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E8:13_1_1_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E4:13_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E10:13_1_1_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>15</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E4:13_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E11:13_1_1_1"/><i><font color="Green">E12</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E8:13_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msinst_1.html#E10:13_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="autalg_1.html#T11">AUTALG_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="msinst_1.html#E8:13_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msinst_1.html#E10:13_1_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E14:13_1_1_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E8:13_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msinst_1.html#E10:13_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="msinst_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E15:13_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E8:13_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msinst_1.html#E10:13_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msinst_1.html#E11:13_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E14:13_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E16:13_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E14:13_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E17:13_1_1_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>thus </b><a NAME="E2:13_1_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="altcat_1.html#V3">associative</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D17">ALTCAT_1:def 17</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D7">ALTCAT_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:13_1_1_2"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<a NAME="E3:13_1_1_2"/><b>then </b><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>20</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font><b> such that </b><br/><a NAME="E5:13_1_1_2"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>18</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>23</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font><b> such that </b><br/><a NAME="E7:13_1_1_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>21</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>26</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font><b> such that </b><br/><a NAME="E9:13_1_1_2"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>24</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E10:13_1_1_2"/><i><font color="Green">E14</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>18</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="autalg_1.html#T11">AUTALG_1:11</a></i>;<br/>
<a NAME="E11:13_1_1_2"/><i><font color="Green">E15</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>21</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="autalg_1.html#T11">AUTALG_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>24</sub></font> <b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E13:13_1_1_2"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:13_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>24</sub></font><b> such that </b><br/><a NAME="E15:13_1_1_2"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msinst_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E16:13_1_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msinst_1.html#E10:13_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msinst_1.html#E15:13_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E17:13_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E15:13_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E18:13_1_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>28</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:13_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E19:13_1_1_2"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:13_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>25</sub></font><b> such that </b><br/><a NAME="E21:13_1_1_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msinst_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E22:13_1_1_2"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msinst_1.html#E11:13_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="msinst_1.html#E21:13_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E23:13_1_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E21:13_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E24:13_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>;<br/>
<b>hence </b><a NAME="E25:13_1_1_2"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E2:13_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msinst_1.html#E13:13_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="msinst_1.html#E15:13_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msinst_1.html#E18:13_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msinst_1.html#E19:13_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msinst_1.html#E23:13_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:13_1_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="altcat_1.html#V5">with_left_units</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D18">ALTCAT_1:def 18</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9">ALTCAT_1:def 9</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:13_1_1_3"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>9</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="msinst_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>9</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font> ;<br/>
<a NAME="E5:13_1_1_3"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T9">MSUALG_3:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E6:13_1_1_3"/>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_1_3"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E5:13_1_1_3"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>assume </b><a NAME="E9:13_1_1_3"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E10:13_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font><b> such that </b><br/><a NAME="E12:13_1_1_3"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>14</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E12:13_1_1_3"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E15:13_1_1_3"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E12:13_1_1_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E16:13_1_1_3"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E12:13_1_1_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_3.html#T4">MSUALG_3:4</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:13_1_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="altcat_1.html#V4">with_right_units</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8">ALTCAT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:13_1_1_4"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>9</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="msinst_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>9</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font> ;<br/>
<a NAME="E5:13_1_1_4"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T9">MSUALG_3:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E6:13_1_1_4"/>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_1_4"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_4"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E5:13_1_1_4"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>assume </b><a NAME="E9:13_1_1_4"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E10:13_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font><b> such that </b><br/><a NAME="E12:13_1_1_4"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>14</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font> <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_4"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E12:13_1_1_4"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E15:13_1_1_4"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msinst_1.html#E12:13_1_1_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E16:13_1_1_4"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" target="_self">MSAlgCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E12:13_1_1_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>;<br/>


</div><b>end;</b></div>


</div>
