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<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>given </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="dtconstr.html#K4">TS</a> <a href="scm_comp.html#K1" target="_self">SCM-AE</a> , <a href="numbers.html#K4">INT</a> <b> such that </b><a NAME="E1:15_1_2"/><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>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="dtconstr.html#NM2">Terminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a>  holds <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="scm_comp.html#K11" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="dtconstr.html#NM3">NonTerminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="scm_comp.html#NM1">bin-term</a><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="bintree1.html#K1">root-label</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="bintree1.html#K1">root-label</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> holds <br/>for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  st <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font> ) )
 ;<br/>

<div><i><font color="Green">E10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>now </b></a><div class="add"><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="dtconstr.html#NM2">Terminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> ;<br/><a NAME="E1:15_1_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="scm_comp.html#K11" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E2:15_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="scm_comp.html#K12" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="scm_comp.html#K11" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="scm_comp.html#K11" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E3:15_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font>)
 <b>by </b><i><a class="txt" href="scm_comp.html#E1:15_1_2"><i><font color="Green">E9</font></i></a></i>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="dtconstr.html#NM3">NonTerminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> ;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="scm_comp.html#NM1">bin-term</a>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> ;<br/><b>assume </b><a NAME="E2:15_1_2_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="bintree1.html#K1">root-label</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="bintree1.html#K1">root-label</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="lang1.html#R1">==&gt;</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> )
 ;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> ;<br/><b>assume </b><a NAME="E3:15_1_2_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/><a NAME="E4:15_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E5:15_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="scm_comp.html#K12" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E6:15_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>)
 <b>by </b><i><a class="txt" href="scm_comp.html#E1:15_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="scm_comp.html#E2:15_1_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scm_comp.html#E3:15_1_2_1"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div>
<b>given </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="dtconstr.html#K4">TS</a> <a href="scm_comp.html#K1" target="_self">SCM-AE</a> , <a href="numbers.html#K4">INT</a> <b> such that </b><a NAME="E3:15_1_2"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="dtconstr.html#NM2">Terminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a>  holds <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="scm_comp.html#K11" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="dtconstr.html#NM3">NonTerminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="scm_comp.html#NM1">bin-term</a><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="bintree1.html#K1">root-label</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="bintree1.html#K1">root-label</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> holds <br/>for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  st <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font> ) )
 ;<br/>

<div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_2"><b>now </b></a><div class="add"><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="dtconstr.html#NM2">Terminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> ;<br/><a NAME="E1:15_1_2_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="scm_comp.html#K11" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E2:15_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="scm_comp.html#K12" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="scm_comp.html#K11" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="scm_comp.html#K11" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E3:15_1_2_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>13</sub></font>)
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:15_1_2"><i><font color="Green">E11</font></i></a></i>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="dtconstr.html#NM3">NonTerminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> ;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="scm_comp.html#NM1">bin-term</a>;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> ;<br/><b>assume </b><a NAME="E2:15_1_2_2"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="bintree1.html#K1">root-label</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="bintree1.html#K1">root-label</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="lang1.html#R1">==&gt;</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> )
 ;<br/><b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> ;<br/><b>assume </b><a NAME="E3:15_1_2_2"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</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> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/><a NAME="E4:15_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E5:15_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="scm_comp.html#K12" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</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="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E6:15_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>)
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:15_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scm_comp.html#E2:15_1_2_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="scm_comp.html#E3:15_1_2_2"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:15_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>from </b><i><a class="ref" href="bintree1.html#S4">BINTREE1:sch 4</a>(<a class="txt" href="scm_comp.html#E2:15_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scm_comp.html#E4:15_1_2"><i><font color="Green">E12</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:15_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E1:15_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="scm_comp.html#E3:15_1_2"><i><font color="Green">E11</font></i></a></i>;<br/>


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