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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">S'</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T'</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green">E44</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</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">S'</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S'</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</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">T'</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T'</font> #) )
 ;<br/>

<b>let </b><font color="Maroon">c</font> be    <a href="waybel_1.html#M1">Connection</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">c'</font> be    <a href="waybel_1.html#M1">Connection</a> of <font color="Maroon">S'</font>,<font color="Maroon">T'</font>;<br/>



<b>assume </b><a NAME="E2:2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<b>given </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>, <font color="Maroon">d</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font><b> such that </b><a NAME="E3:2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="waybel_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">d</font> is <a href="orders_3.html#V5">monotone</a> )
 <b>and </b><br/><a NAME="E5:2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><br/> for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">t</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> iff <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">s</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D10">WAYBEL_1:def 10</a><br/>

<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S'</font>,<font color="Maroon">T'</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">d'</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T'</font>,<font color="Maroon">S'</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">g'</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D10">WAYBEL_1:def 10</a><br/>

<b>take </b>
<font color="Maroon">d'</font>
;<br/>

<b>thus </b><a NAME="E8:2"/>
<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">d'</font></span><a href="waybel_1.html#K1">]</a></span>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E9:2"/>
( <font color="Maroon">g'</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">d'</font> is <a href="orders_3.html#V5">monotone</a> )
 <b>by </b><i>, , <a class="ref" href="waybel_9.html#T1">WAYBEL_9:1</a></i>;<br/>

<b>let </b><font color="Maroon">t'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T'</font>;<br/><b>let </b><font color="Maroon">s'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S'</font>;<br/>



<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E12:2"/>
( ( <font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s'</font> implies <font color="Maroon">t</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> ) &amp; ( <font color="Maroon">t</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> implies <font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s'</font> ) &amp; ( <font color="Maroon">d'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s'</font> implies <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> ) &amp; ( <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> implies <font color="Maroon">d'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s'</font> ) )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>hence </b><a NAME="E13:2"/>
( (  not <font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s'</font> or <font color="Maroon">d'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s'</font> ) &amp; (  not <font color="Maroon">d'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s'</font> or <font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s'</font> ) )
 <b>by </b><i/>;<br/>


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