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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">N1</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>, <font color="Maroon">N2</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>, <font color="Maroon">N3</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/>



<b>given </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">N1</font>,<font color="Maroon">N2</font><b> such that </b><a NAME="E1:58"/><i><font color="Green">E29</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N1</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N2</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N1</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N1</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">p</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12" target="_self">YELLOW_6:def 12</a><br/>

<b>given </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">N2</font>,<font color="Maroon">N3</font><b> such that </b><a NAME="E3:58"/><i><font color="Green">E41</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N2</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N3</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:58"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N3</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N2</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N2</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">p</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12" target="_self">YELLOW_6:def 12</a><br/>

<b>take </b>
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12" target="_self">YELLOW_6:def 12</a><br/>

<b>thus </b><a NAME="E5:58"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N1</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N3</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_6.html#D1" target="_self">Def1</a>, <a class="txt" href="yellow_6.html#E2:58"><i><font color="Green">E32</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>

<b>let </b><font color="Maroon">m</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N3</font>;<br/>

<b>consider </b><font color="Maroon">m1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N2</font><b> such that </b><br/><a NAME="E7:58"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N2</font>  st <font color="Maroon">m1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Maroon">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N1</font><b> such that </b><br/><a NAME="E9:58"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N1</font>  st <font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Maroon">m1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N1</font>;<br/>

<b>assume </b><a NAME="E10:58"/>
<font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E11:58"/><b>then </b>
<font color="Maroon">m1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:58"/><b>then </b>
<font color="Maroon">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E13:58"/>
<font color="Maroon">m</font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


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