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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">N</font> holds <br/> ( <font color="Olive" title="b2">M</font> is <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a> &amp;  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Olive" title="b2">M</font> )</span><br/><b>let </b><font color="Maroon" title="c2">N</font> be  <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( <font color="Olive" title="b1">M</font> is <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a> &amp;  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Olive" title="b1">M</font> )</span><br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font> is <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a> &amp;  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c3">M</font> )</span><br/>





<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E2:24"/><i><font color="Green" title="E17">A1</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E3:24"/>
for <font color="Olive" title="b1">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font> st <br/>for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b3">p</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12" title="YELLOW_6:def.12">YELLOW_6:def 12</a></i>;<br/>
<a NAME="E4:24"/><i><font color="Green" title="E18">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:24"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:24"/>
for <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">n2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b2">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> holds <br/>the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n1</font>, <font color="Maroon" title="c6">n2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c6">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> implies the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24_1"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c5">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font>
 <b>and </b><br/><a NAME="E2:24_1"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c6">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n2</font></span><br/>

<a NAME="E3:24_1"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="#E4:24"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E1:24_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:24_1"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="#E4:24"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E2:24_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b>the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> = 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E4:24"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E1:24_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:24"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E3:24_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="#E4:24_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n2</font>
<b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E4:24"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E2:24_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:24"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E8:24"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c3">M</font> is <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a>
 <b>by </b><i><a class="ref" href="yellow_6.html#D6" title="YELLOW_6:def.6">YELLOW_6:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c3">M</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:24"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E11:24"/><i><font color="Green" title="E23">A11</font></i>: 
 <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_value_of</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D1" title="YELLOW_6:def.1">YELLOW_6:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font>;<br/>
<i><font color="Green" title="E24">A12</font></i>:  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_value_of</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> = 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:24"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E10:24"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="#E11:24"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>
<b>thus </b> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c2">N</font> = 
 <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_value_of</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="ref" href="yellow_6.html#D10" title="YELLOW_6:def.10">YELLOW_6:def 10</a></i>
<br/>.= 
 <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_value_of</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font>
<b>by </b><i><a class="txt" href="#E7:24"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="#E13:24"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="yellow_6.html#D1" title="YELLOW_6:def.1">YELLOW_6:def 1</a></i>
<br/>.= 
 <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c3">M</font>
<b>by </b><i><a class="txt" href="#E8:24"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="yellow_6.html#D10" title="YELLOW_6:def.10">YELLOW_6:def 10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
