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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green">E8</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="waybel_0.html#NM2">net</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="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:12"/><i><font color="Green">E9</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #)
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green">E10</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="waybel_0.html#V6">strict</a>  <a href="waybel_0.html#L1">NetStr</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:12"/><i><font color="Green">E11</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>5</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>6</sub></font> #)
 <b>and </b><br/><a NAME="E6:12"/><i><font color="Green">E12</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel33.html#E1:12"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel33.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E8:12"/><i><font color="Green">E13</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E9:12"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>
<a NAME="E10:12"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a></i>;<br/>
<a NAME="E11:12"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="waybel_0.html#T3">WAYBEL_0:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a>, <a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a>, <a class="ref" href="waybel_8.html#T13">WAYBEL_8:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="waybel33.html#E2:12"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel33.html#E5:12"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E14:12"/>
<font color="Maroon">c<sub>8</sub></font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a><br/>

<b>thus </b><a NAME="E1:12_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel33.html#E3:12"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel33.html#E6:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel33.html#E8:12"><i><font color="Green">E13</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel33.html#E2:12"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel33.html#E9:12"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:12_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E8:12_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<a NAME="E9:12_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E4:12_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:12_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="waybel33.html#E2:12"><i><font color="Green">E9</font></i></a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:12"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="waybel_0.html#V6">strict</a>  <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>5</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Olive">b<sub>1</sub></font> #) &amp; the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel33.html#E6:12"><i><font color="Green">E12</font></i></a></i>;<br/>


</div>
